Math 122.C / Fall 2021

Homework

The "assigned" dates for HW on this website are approximate.
The rule as to when each HW is actually assigned is stated in the Syllabus:
HW for a section is assigned on the day when we have finished covering that section in class.


Note about the prerequisite material from Calculus II
The material that you must know from Calculus II is:
  1. Definition of vectors in 2D and 3D;
  2. Geometric addition and scalar multiplication of vectors (this includes parallel vectors);
  3. Equations of lines in 2D and of lines and planes in 3D.
If you do not remember some of this material, you must review it on your own.
- Topics 1 and 2 are reviewed in Sec. 12.2 of your Calculus textbook by Stewart,
and also in Secs. 2.1 and 2.2 of your Linear Algebra textbook. (You should focus
only on these topics in Secs. 2.1 and 2.2 and disregard other topics there.)
- Topic 3 will be very briefly reviewed in Sec. 3.1.
It is also reviewed in Sec. 12.5 of Stewart's Calculus and in Sec. 2.4 of your Linear Algebra book.


HW # 1

Assigned on:  09/01
Sec. 1.1:  ##  1, 5, 6, 9, 11, 15, 21, 29,   31, 33, 35.
                Answer for # 6:  linear.
                Note for ## 31, 33, 35:    You are not asked to solve these linear systems
                                                      (because we haven't yet learned how to do so).
                                                      You just need to do the step described in the paragraph before # 30.


HW # 2
Assigned on:  09/08
Sec. 1.2:  ##  3, 5, 7, 9, 10, 13, 15, 17, 21, 23, 27, 29, 31, 37, 41.
             
Clarification for ## 3 -- 10: Do either (a) or (b), whichever applies.
                                                      You are encouraged to check you answer with MATLAB.
                                                       For relevant MATLAB's syntax, see Appendices A.1 -- A.3
                                                       and Ex. 5 in Sec. 1.2.
                                                       Alternatively, you may use:
                                                       - The two online REF calculators, the links to which are given
                                                          under the link for Notes for Sec. 1.2;
                                                       - Mathematica (in its Help, enter "reduced echelon form" to find out
                                                         the name of the correct command; just "ref" won't work).
              Clarification for ## 13 -- 21:   You need to first put the augmented matrix in REF.
              Hint for # 41:   This is a linear system for the cos(alpha) and sin(beta).

Extra credit # 1  (the weight of each problem, added to the final grade, is shown next to the problem)
Assigned on:  02/08,   Due:  02/15
Before you attempt this extra-credit assignment, please read these instructions!
Sec. 1.2:  ##   49 (0.1%), 51 (0.1%), 56
(0.1%).
Note:  In ## 49, 51, provide solutions with complete explanations.
          I.e., you need to explain both the details of the setup and how you solved
          the l.s. via reduction to the REF.
          No credit
will be given if:
             - You miss any details in the setup that I deem essential
               (
i.e., brief solutions, easily found online or in the Solution Manual, will receive no credit);
             - You do not use the REF.) If you used Matlab for that, you need to attach a printout.


HW # 3
Assigned on:  09/12
Sec.
1.3:  ##  1, 3, 5, 6, 19, 21, 25, 27, 29, 32, 33.
                    Answer for # 6:   no (find a statement in the Notes or Book from which this follows directly).
                    Clarification for # 25(b):
                                         The matrix B here is the same as defined previously in (a). 
                    Note for ## 27, 29, 32, 33: 
                                         Make sure to follow all steps of the Example presented in class (and in the
                                         posted Notes, as well as the corresponding must-read textbook Example).
                    Note for ## 29, 32, 33:  I recommend that you obtain the REF of the respective matrices
                                                       using either MATLAB or the other resources mentioned in the
                                                       "Clarification for # 3" in HW # 2 above.
                                                       If you use the rref command in Matlab, then before you do so,
                                                       execute the comand
                                                       format rat
                                                       This will prevent Matlab from converting rational fractions into decimals.
                    Answer for 32:   x^2 + y^2 - 3x -5y + 6 = 0
                    Answer for 33:   7x^2 + 7y^2 - 39x - 23y + 50 = 0
                            


HW # 4
Assigned on:  09/15
Sec. 1.5:  ##  5, 9, 11,
21, 33, 3529, 31, 53, 55, 59;  43, 45, 47;  
                    40, 50; 
52(d), 13, 15, 17, 61(a,b);  65(a,b).
p. 106, # 6(a).

                    Note for # 21:    See a similar Ex. 5 in Sec. 1.5 of the textbook.
                    Note for ## 43, 45, 47:   These are based on Ex. 2, 3 in the book,
                                                         which you were supposed to read on your own.

                    Note for ## 40, 52(d):     To check your answer, use Matlab.
                    Hint for # 50:       These numbers are 8 and 12, but not necessarily in that order.
                    Note for # 61(a):   This is asking you to find the MATRIX FORM of the linear system.
                                                Recall that the "matrix form" is a must-know concept, which
                                                you were required to learn by reading Example 5 in the textbook
                                                and a Clarification about it on p. 4-12 of the (separately) posted Note.
                    Note for # 61(b) and p. 106, # 6(a):
                                                   Use the formula that should be prominently displayed
                                                   on the front cover of your notebook.


Note about the prerequisite material from Calculus II
As noted at the top of this page, the material that you must know from Calculus II is:
  1. Definition of vectors in 2D and 3D;
  2. Geometric addition and scalar multiplication of vectors (this includes parallel vectors);
  3. Equations of lines in 2D and of lines and planes in 3D.
If you do not remember some of this material, you must review it on your own.
More specifically, in Sec. 1.7 we will begin actively using the material of Topics 1 and 2.
Review them now, or else you may find yourself floundering when related material is
covered in class, as well as when related questions are asked on quizzes and tests.
- Topics 1 and 2 are reviewed in Sec. 12.2 of your Calculus textbook by Stewart,
and also in Secs. 2.1 and 2.2 of your Linear Algebra textbook. (You should focus
only on these topics in Secs. 2.1 and 2.2 and disregard other topics there.)


HW # 5
Assigned on:  09/17
Sec. 1.6:  ##  7, 13, 15, 17, 19, 21, 26;
Sec. 1.5:  # 65(c);
Sec. 1.6:  ##
28, 27 [yes, in this order], 31, 35, 41(a), 43, 47, 49, 57.
                   
                    Note for # 26:  Do this for general n x n matrices, not for 2 x 2.
                                           Start by distributing the product on the left-hand side.
                                           The entire solution should take at most two short lines.
                    Notes for # 65(c) of Sec. 1.5:                                 
                                           1) The reason this problem is assigned here is because it is related to
                                                the next problem (#28).
                                           2) You still need to solve it using the same method by which you
                                               solved # 65(a,b) in Sec. 1.5.
                                           3) The "a" and "b" in the answer to this problem at the end of the book
                                               both occur from the "x2 = free" in the solution of respective linear systems.
                                               "a" represent "x2" in the first linear system that you have solved,
                                               and "b" represents "x2" in the other linear system.
                                               The reason that one used different "a" and "b" is that the "x2" is
                                               different in these unrelated linear systems.
                                           4) Now, find a statement at the beginning of Sec. 1.6 that states the ge
neral fact
                                               illustrated by # 65(c). This should help you do #28.
                    Note for # 28:  The point of this problem being assigned is that you relate it
                                           to an example done in class, where we saw that certain
                                           cancellations, which you can do for scalars,
                                          
do not, in general, work for matrices.
                    Note for # 27:   Relate this problem to what you did for # 28 and what we did in class.
                    Note for # 31:  # 30 was done in class.
                    Note for # 35:  The "outer product", referred to in the problem statement, is just a name.
                                           (You are not required to memorize it; we'll briefly encounter it again
                                             at the end of the course.) I surmize that the reason why the book uses
                                            it here is to draw your attention to the fact that this is not the same as the
                                            scalar product (also called inner product), defined earlier in this section,
                                            even though the inner and outer products look similar.
                    Hint for # 41(a):  Write this as a linear system.
                    Hint for # 43(b):  The crux of this (admittedly, very special) problem is that
                                              not only should you not form A^5, but you should
                                              also not do any additional matrix-vector multiplications!
                                              Instead, use the following guidelines.
                                              1) Do this problem for A^2 u . Use 43(a) and part 1 of
                                              Thm. 8  where  C = u. Now use part 3 of Thm. 8.
                                              This should give you the answer for A^2 u .
                                              2) You should now be able to generalize it for A^5 u .

                                                              If you still have difficulties, repeat the previous steps for A^3 u.
                     Note for # 49:  One part of this problem was done in class.

Extra credit # 2  (each problem is worth 0.2% added to the final grade)
Assigned on:  02/17        Due02/24
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 1.6:  # 46  + 
Sec. 1.5:  # 67.
(The regular assignment that you need to turn in with this Extra Credit is only for Sec. 1.6.)
                     Note for # 67:  B is  n x n,  not  m x n,  as the book says.
                                           Also, you must show all details of your work to get credit.


HW # 6
Note about the prerequisite material from Calculus II
Before attempting this assignment, please read the Note about the prerequisite material
posted before HW # 5, and do as it directs.
Assigned on:  09/
Sec. 1.7:  ##  1, 3, 5, 6, 56, 9, 11, 13;  29, 30
[Answer: a=1], 33;  41, 45, 47;
                   
17, 19, 20, 21, 48, 35, 37, 39, 53, 54.
                    See also the additional assignment  and  Word Problems  below.

                    Notes for ## 5, 6, 47(a):   1) Find the answer in the Notes (or in the book).
                                                          2) The best solution will not require any calculations.
                    Note for # 56:  Ignore the Hint from the book. Focus on the geometric
                                          solution of this problem, as explained in the Notes.
                    Note for ## 41, 45, 47:
                                           - One way to find a given  b  as a linear combination of v1 and v2
                                           would be similar to what we did on p. 6-2 of the posted Notes:
                                           set up an equation asking if  v1, v2, b  are linearly dependent or not
                                           and then, after finding c1, c2, c3, solve for  b  as we solved for  v3.
                                           (We also followed this method on p. 6-11.)
                                           - Another way is more direct:
                                           Set up the equation  k1*v1 + k2*v2 = b  and convert it to a linear system
                                           using the Key Formula. (Compare this with the last equation on p. 6-11.)
                    Note for # 20:  Find the answer using your answer for # 17 and
                                          one of the examples of proofs in the Notes.
                                          BTW, this should jive with your answer for # 19, too.

                    Note for # 48:  If the Hint in the book doesn't help by itself, try to
                                           relate it to the solution you have found in # 21.
                                           (There, of course,  v3 = 0 and not v1.)
                    Note for # 53:  # 52 was done in class. You just need to relate that
                                          class example to # 53 in the same way as ## 27 and 28
                                          in Sec. 1.6 were related (which was also done in class
                                          when we covered Sec. 1.6).
                    Hint for # 54:  For one of the columns of B, use the formula that should be
                                          prominently displayed on the cover of your notebook.
                                          Then use one of the Theorems about a (non)singular matrix.

In addition, you must do the following:
                    For each of ## 1, 3, 6, 9, 11, 29, 30, 47(a) -
                          - Make a sketch supporting your conclusion about linear dependence or
                          independence of the vectors. (When sketching vectors in 3D, simply indicate
                          whether they are in the same plane or not.) Refer to the posted p. 6-7 (the figure
                          called "parallelogram") for
how to illustrate that any 3 vectors in the same plane
                          are linearly dependent.

                          You should also make use of the conclusions of Ex. 4 and Thm. 11 when
                          doing some of these problems.
                    For each of ## 17, 19, 20 -
                          - Sketch the vectors representing the columns of the matrix, so that it
                          is clear whether the vectors are linearly dependent or independent.
                          (Consult Theorem 12 for the answer, and the posted p. 6-8 for a sketch.)
                    For each of ## 35, 37, 41, 45, 47(b) -
                          - Make a sketch illustrating that the r.h.s. is indeed a linear combination of
                          the columns of the matrix. Your sketch must show the vectors representing
                          the columns of the matrix and the components x1 and x2 of the solution x .
                          Again, consult the figure "parallelogram" on the posted p. 6-7.
      I also strongly suggest that you use the Matlab file posted under Notes for Sec. 1.7-A.

Additional Word Problems for HW 6 / Sec. 1.7:
(They are posted here for a good reason; so - don't skip them!)
Word Problem 1: (a)  The columns of square matrix A are linearly independent. How many solutions
                                does the linear system   A x = 0  have?
                          (b)  Same question is the columns of A are linearly dependent.
                          (c)  Matrix A is the same as in (b). What are the possibilities for the
                                number of solutions of 
A x = y  where y does not equal 0 ?
                          Hint:  Answers to all these questions are found by combining at most
                                   two theorems from the book and pp. 6-13, 6-14 of the posted notes. 
Word Problem 2: (i)  Use Matlab's command  rref  (see Ex. 5 in Sec. 1.2 of the book and Appendices A.1 -- A.3)
                                 to solve the following 3-by-3 linear systems  
A x = y , where A and y are given below.
                           (ii) 
Use your results and the statements from p. 6-14 to determine whether
                                 matrix  A  above is singular or nonsingular in each of the cases.
                           (iii) Which of the cases corresponds to which case in Word Problem 1?

                           (a)         ( 1  4  7 )              ( 10 )
                                 A = ( 2  5  8  ) ,     y =  ( 11 )  ;
                                        ( 3  6  9 )              ( 12 )
                           (b)  Same as in (a), but  "9" in A and "10" in  y  are interchanged;
                           (c)  
Same as in (a), but  "10" and "11" in  y  are interchanged.

                           Answer to (iii):  (b, a, c).
                          
                    

Extra credit # 3  (problem weights, to be added to the final grade, are indicated below)
Assigned on:  09/24           Due:   10/03
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 1.7:  ##   57
(0.2%), 58 (0.15%), 59 (0.15%).


HW # 7
Assigned on:  09/24
Sec. 1.8:  ##  1, 5, 27, 7.


HW # 8
Assigned on:  10/01
Sec. 1.9:  ##   1, 3, 5, 7,  13, 17, 19, 23, 25, 27, 29, 35, 37, 38, 39, 41, 43, 45, 49, 51,
                     55 [you do not need any results from #54; just use the notes], 67, 68, 69;
p. 107, # 18.

Word Problem:    Let A, B, C be (n x n) invertible matrices. Simplify:
                          (A B)^(-1) * (C B^T)^T * (A^T C^(-1))^T.  

                     Notes for # 27:   1) There is a typo in the book: the answer should be
                                                  " lambda is NOT equal to 2 or -2".
                                      2) Do not use the method from the book which refers to the
                                                  quantity "Delta" (looking like a triangle). Instead,
                                                  you should use Thm. 18 to relate the desired property
                                                  of  A  to another property which you have learned earlier.
                                                  Then use one of the methods (also studied earlier)
                                                  to investigate the latter property of  A.
                     Notes for ##
35, 37, 38, 39, 41, 43, 45:
                                                1) The formula "(9)", which the statement of these problems
                                                     refers to (so inconspucuously!) is fond immediately above
                                                     that statement.
                                                2) In these problems, you must not compute any inverses!
                                                    Instead,  use Theorem 17.
                     Answer for # 38:   C^(-1)*(A^(-1))^T.
                     Hint for ## 55, 69:  Review steps of one of the proofs done in class.
                                                  Remember to begin your proof by writing what is given
                                                  and what you want to show.
                     Hint for # 68:  Use Thm. 17.
                     Hint for # 69:  Use the same method as in # 55.
                                           Review the class Example mentioned in the Hint for that problem.
                     Answer for # 18 on p. 107:   B^(-2).
                     Answer for Word Problem:   B^(-1) A^(-1) B A

Extra credit # 4   (problem weights, to be added to the final grade, are indicated below)
Assigned on:  
          Due:  
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 1.9:  ##    56 (0.2%), 52 (yes, in this order)
((a-c)=0.15%, (d)=0.35%),
                      74
((a)=0.1%, (b)=0.2%), 66 (see below) (0.2%).

                      Note for # 52(d):  You must begin by explaining, in general terms, which step of the
                                                 "derivation" implied in (a) fails to imply that X=I or X=I.
                                                 (See the Hint below).
                                                 Then, for each of the specific matrices (A in (b) and B in (c)),
                                                 you must show *explicitly* why (X-I)(X+I)=O and yet X-I and X+I
                                                 are both different from O.
                       Hint for # 52(d): 
See # 28 in Sec. 1.6 and find a very similar statement in
                                                 the notes for Sec. 1.6 (it is in the topic "Differences from
                                                 scalar multiplication"). 
Also, there is another related problem
                                                 in the regular part of this assignment.
                      Note for # 74:    # 72, mentioned here, was proved in class (Ex. 5 in posted Notes for Sec. 1.9).
                      Note for # 66:    The difference between two solutions for the vectors b1 and b2
                                               given in the book is not really that impressive. So, to see a
                                               more impressive case, take
                                               c1=[1 1 1]^T and c2=[1.01 1.01 0.99]^T,  or better yet,
                                               d1=[1 1.5 1]^T and d2=[1 1.51 0.99]^T.
                                               Also, compute the largest percentage difference between
                                               the entries of solution  x  for one pair of the right-hand-side vectors
                                               (e.g., max_{i=1...3} (xc1(i) - xc2(i) )/xc1(i) , where
                                                xc1 is the solution with the r.h.s. c1 and xc1(i) is its i-th entry).
                                   
              Here and everywhere below,
            the notation    [a b c]^T     means

            the colum vector [a]
                                         [b]
                                         [c] .
             So, "^T" stands for "the transpose" of the row vector it is applied to.

             As you can tell from this example, it is much easier for me
             to use the "^T" notation
to transpose a row-vector and
             hence make it a column than to type a colum vector.


Note about the prerequisite material from Calculus II
Before attempting this assignment, please read the Note about the prerequisite material
posted at the top of this page (before HW # 1), and do as it directs.


HW # 9
Assigned on:  03/03
Sec. 3.1:  ##   5, 7, 13, 14, 15, 16, 18, 19, 20, 23, 25, 27, 28, 29, 30.
                     Note for ## 5, 7:  We first sketched the sum of two vectors in Sec. 1.7.
                                               If you forgot how to do so, read Sec. 2.1 - all subsections
                                               about vector addition, scalar multiplication, and subtraction.
                                               This material is your prerequisite material from Calculus II.
                     Note for ## 19:   You do not need to make a sketch. Instead, characterize this geometric object
                                               by stating two facts about it, as discussed in class.
                     Answers:   # 14 - line x=0;  
                                     # 16 - line y=3x;  
                                     # 18 - positive part of x-axis;
                                     # 20 - a line along [2 0 1]^T; (see the green text above)
                                     # 28 - t*[2 -3 1] for all t; 
                                     # 30 - [x1, 2, x3] for all x1, x3.
Sec. 1.1:  ##   16, 17, 18.  (Yes, this is not a typo: Sec. One point one.)
                     Suggestion:  Use the rref command in Matlab.
                     Note:   The free variable, which in these problems will be  x3,  was denoted  t  in Vector Calculus.
                     Answers:   # 16 - planes intersect; line x1 = -t/2 +1/2,  x2 = 0*t + 2, x3 = t + 0;
                                     # 18 - planes coincide.

                     Word Problems
                     Interpret the subset W in R^3 geometrically (a.k.a. "give a geometric description of"):
                     1.   [x1,x2,x3]^T where  x3=5*x2.
                     2.   [x1,x2,x3]^T where  x1+x2=0 and x1+x2-4*x3=0.
                     Hint:   See both examples done in class.


HW # 10
Assigned on:  03/08
Sec. 3.2:  ##   1, 3, 5, 6 [a subspace, but not proper], 7, 9, 10, 12, 13, 14,  15, 16, 17, 19, 23, 21, 27.
                     Word Problems
                     Interpret the subset W in R^3 geometrically by sketching the graph of W
                      and then prove that W is not a subspace of R^3:
                    1. In # 15 of the main part of this assignment, replace the comma (which means "and")
                        by "or".
                    2. In # 21, replace "and" by "or".

              General Note 1:  The phrase "give a geometric description" does NOT imply sketching the object.
                                       Instead, you should describe the object in words and state two of its key properties,
                                       as was done in examples considered in class.
              General Note 2:  Before doing this HW, review both examples done in class for Sec. 3.1.
              General Note 3:  Some problems define W using a notation like this:  x1 = x2 = 0.
                                       Note that this means two separate equations: x1 = 0 and x2 = 0
                                       (or, equivalently, x1 = x2 and x2 = 0).

              Hints for # 3 and both Word Problems
                           1)  See the must-read Example 4 in Sec. 3.2 of the textbook.
                                It explains that the word "or" defines a union of the two lines
                                (which includes all points of both lines).
                                This should be contrasted with the meaning of the word "and",
                                which stands for the intersection of the two lines (which includes
                                only the one point common to both lines). See also Example 2 in the
                                posted Notes for Sec. 3.1 (not 3.2).
                            2) Begin by plotting W, following the description in the aforementioned Example 4.
                                Then try to give a geometric proof. (A similar proof was done in Example 6 in
                                the Notes for Sec. 3.2.)

               Answer for # 10:   yes; plane perpendicular to <1,-1,1> and going through the origin.
 
              Answer for # 12:   yes; plane perpendicular to <1,0,-2> and going through the origin.
               Answer for # 14:   yes; the (x1,x3)-plane (it goes through the origin, obviously).
               Note for # 15:       If you denote x3=t, you will get the answer in the back of the book.
               Answer for # 16:   yes; line at the intersection of planes 2*x1+0*x2-1*x3=0 and 0*x1+1*x2-1*x3=0;
                                          since each plane goes through thre origin, so does the line.
               Note for # 27:       The purpose of this Exercise is to make you actually look at the properties
                                          on p. 168, as was repeatedly said in class.



HW # 11
Assigned on:  03/
Sec. 3.3:  ##   1, 5, 6, 9, 13, 15, 17, 19, 22, 25, 27, 29, 31, 32, 35, 39, 41, 42, 43, 45, 50.
                     Answer for # 6:   {xx=[a, -a]^T for all a}; line x2 = - x1.
                     Note for ## 15, 17, 19:  
                                             Use Ex. 1 from the textbook.
                                             When working through this Example, you will need to recall two facts
                                             to understand the final steps of this example:
                                              - How does the augmented matrix tell you if the linear system
                                                is consistent or inconsistent? See Example 2 in posted Notes for Sec. 1.2.
                                                (This should make it clear why the book required the condition
                                                  (1/2)*y3 + (1/2)*y1 - y2 = 0.)
                                              - What is the general form of the equation of a plane in 3D?
                                                (See posted Notes for Sec. 3.1.)
                     Note for # 22:   Use the definition of N(A); answers: a, c, e.
                     Note for ## 27 - 35:  You may determine R(A) using the method of Example 1 of the book.
                                                    However, a much easier way is to base your solution on Examples 1 - 3
                                                    of the Notes. There, you will first need to answer the problem's question
                                                    geometrically, and then based on the geometric answer, write the
                                                    algebraic specification of R(A) interpreting it as the span of columns of A.
                     Answer for # 32:   N(A) = {[0 0]^T};
                                                 R(A) = {y:  y=[y1,y2,y3]^T where  3*y1 -2*y2+y3 = 0},
                                                 or, equivalently,
                                                 R(A) = {y:  y=c1*[1 2 3]^T + c2*[3 7 5]^T }.
                     Answer for # 42:   [2 -3 1; -1 4 -2;  2 1 4];  it is a subspace of R^3 since so is any R(A).
                     Answer for # 43:   A = [3  -4  2].
                     Answer for # 50:   (a) R=R^n, N=0;   (b) R=0, N=R^n;   (c) R=R^n, N=0.


HW # 12
Assigned on:  03/
Sec. 3.4:  ##   32 [yes], 33, 34, 35, 27, 21(a), 23(a), 1, 3, 5, 6, 7, 9, 11(a--c), 15(a--c), 25. 

                    Additional assignment for # 27:   Find the coordinates of  v3  in the basis  {v1,v2}.
                    Additional assignment for # 23(a):   Find the coordinates of the the linearly dependent
                                                                         vectors in set S in the basis you have found.

                     Note for ## 1,3,5,6,7:   Recognize these problems as finding a basis of the nullspace of
                                                       some matrix. Then use the corresponding Method of finding a
                                                       basis from the Notes (and the Must-Read Example listed for that
                                                       Method).  You may also find
useful Ex. 2 in the book for Sec. 3.3.
                     Additional assignment for # 9:    I
f x is in W, state the coordinates of x in this basis.
                     Additional assignment for ## 11(c) and 15(c):    
                                    For each Aj not in the basis of R(A), find the coordinates of Aj in this basis.

                     Answer for # 34:  Will not be posted. Reason: It can be found without calculation, based
                                               on the definition of a basis and on one of the Theorems in Sec. 1.7.
                     Answer for the additional part of # 27:    (2,3).
                     Answers for the additional part of # 23(a):
                                   coordinates of S3 and S4 in the basis {S1,S2} are (1,1) and (3,-1).
                     Answer for # 6:   { [2 2 1 1]^T }.
                     Answers for the additional part of # 9:     (a) (2,1);  (c) (0,-3);  (d) (2,0).
                     Answers for the additional part of ## 11(c) & 15(c):    
                                   # 11(c):  coordinates of A3 and A4 in the basis {A1,A2} are (1,1) and (1,-1).
                                   # 15(c):  coordinates of A2 in the basis {A1,A3} are (2,0).
 

HW # 13
Assigned on:  03/
Sec. 3.5:  ##   3, 5, 9, 11, 13, 15, 17, 19, 20, 29, 25, 27, 33, 35, 39.
                     Note for ## 20, 29:
                                           The augmented matrix corresponding to the given set of
                                            equations must first be put into REF, as in one of the
                                            examples in the Notes.
                                            Remember that in this course, you must never solve a system of equations
                                            by manipulating their terms. You must always use the REF algorithm!
                     Answer for # 20:    dim(W) = 1.
                     Note for ## 25, 27:
                                              The answers in the book are correct, but I am not quite sure
                                           how
they got them. Most likely, they used Strategy 2 of finding a basis
                                           for the range of a matrix. However, recall that this Strategy requires
                                           rather cumbersome calculations: You would first need to find
                                           the equation for R(A) as in Ex. 1 of Sec. 3.3/book, which is tedious, and
                                           then follow the steps of Method 2 (that for finding a basis for the null space).
                                              In contrast, Strategy 1 is much simpler to use.
                                           With Strategy 1, I get the following answers:

                                           # 25:  the basis is {A1, A2};
                                           # 27(a), basis is {v1, v2, v4};
                                          
# 27(b), basis is {v1, v2, v3}.
                                              This illustrates that you should use Strategy 2 for finding R(A) only when
                                           your R(A) is *already* given in the form as in Ex. 5 of Sec. 3.4.
                                              In other cases, when finding a basis for R(A), you should use Strategy 1.
                     Hint for # 35:  Find a related Theorem in the Notes (the same Theorem is
                                           stated in the book somewhat differently, so it is not as easy
                                           to recognize its connection to this problem).


HW # 14
Assigned on:  03/
Sec. 3.6:  ##   3, 5, 6, 9, 10, 11, 15, 17, 21, 28 [Hint:  ||v||^2 = v^T * v].
                     Note for ## 5,6:  Make sure that you are finding an orthogonal set, not just a
                                              linearly independent one. (Obviously, use the same condition of
                                              a set being orthogonal as in # 3. If you forgot what "orthogonal" means,
                                              find a definition in the posted Notes or in the book.)
                     Answer for # 6:   a=-c/2,  b=5/2*c,  c is arbitrary.
                     Note for ## 9, 10, 11:   Note that you are asked to find coordinates of given vectors
                                                     in given orthogonal bases, as was shown in classThus,
                                                     the method you must use is different from that used in Sec. 3.4.
                     Answer for # 10:   (1, 1, 0).
                     Note for # 28:       It will suffice if you do this problem for p = 3.

Extra credit # 5   (each problem is worth 0.2% added to the final grade)
Assigned on:  03/           Due:   03/
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 3.6:  ##   22, 24, 25, 26.


HW # 15
Assigned on:  03/
Sec. 3.7:  ##   13, 16, 17, 3, 5, 7, 19, 20, 25, 29, 30, 33, 39, 43, 44 [Answer:  a*I].
                    Word Problem:
                     Find the matrix P of projection onto the direction of the vector w=[1; 2].
                     What vector  x  satisfies  P x = 0?  Try to answer this geometrically, then
                     read off coordinates of  x  from your figure.

                     Note for ## 13, 16, 17:  First, try to use the Claim from p. 15-3 of the posted Lecture Notes.
                                                        When you cannot come up with the corresponding A, check the
                                                        properties of a linear transformation.
                     Answer for # 16:   Yes; see Ex. 5(a) in the notes.
                     Note for ## 3, 5, 7:   Find the matrix of the transformation and restate the
                                                    question of the problem in terms of the range or
                                                     null space of that matrix.
                     Note for ## 19, 20:  Do NOT use a method where you need to first find
                                                   A = U * V^{-1}. This is not wrong but is overly complicated.
                                                   Instead, use the method from a very similar Example done in class.
                     Answer for # 20:   (a) [-3; 3];   (b) [-6; 0];  (c) [-9; 7].
                     Answer for # 30:   [2  -1  4].
                    
Note for # 39:   The notation  G o F  is just a name given to the two-stage
                                           transformation G(F[u]). It might have as well be called  H[u],
                                           or whatever.  All that is relevant is that first, u is transformed by F,
                                           and then the result,  F[u], is transformed by G. 
                     Hint for # 39:      - One method is to verify that G(F[u]) satisfies the two
                                           properties of a linear transformation. (E.g., the first property
                                           would be  G(F[x+y]) = G(F[x]) + G(F[y]),  and similarly for the second one.) 
                                           For each property, the verification that you need to do is a two-step process.
                                           In the first step, you will need to use the linearity of F, e.g., F[x+y] = F[x] + F[y].
                                           Then in the second step, you will similarly use the linearity of G.
                                              - An alternative (and shorter!) method is to use the relation
                                           between linear transformations and matrices.


Extra credit # 6   (each problem is worth 0.2% added to the final grade)
Assigned on:   04/        Due:  04/
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 3.7:  ##   47, 48, 49.
                     Word Problem:   Use the formula that relates the matrix of a transformation
                                               to the results that the transformation has on basis vectors
                                               to obtain the matrix of reflection about the line
                                               y = tan(alpha) * x.
                                               (This will be the same matrix as you were asked to derive
                                                 in the Bonus problem for Exercise 1 in Project 1.)
                      Note : You may choose any basis that you find the most convenient in this case.


HW # 16
Assigned on:  04/
Sec. 3.8:  ##   1, 5, 7, 9, 3, 11, 13, 16 [use Generalization 2].
                     Note for # 9:   There seems to be a typo in the book. The correct answer is  y=0.6+1.3t.
                     Notes for # 3: 1.  Your normal equation will end up being a 3x3 linear system with
                                                  unpleasant-looking entries. Solve it by Matlab or Mathematica.
                                             2. The point of this problem is to demonstrate what happens when A is
                                                  singular.
As you may remember from Sec. 1.7, then  B=A^T*A  is also
                                                  singular.
Then, we showed that if  B*x = b  with a singular  B  is
                                                  consistent, then
it has infinitely many solutions. This is what you
                                                  should find in this probhlem.

                     Note for # 16(a):   This simply asks you to set up the inconsistent linear system  A*x = b,
                                                 as was done in class for the linear fit.
                     Answer for # 16(b):   a1 = 69/58,  a2 = 9/58

Extra credit # 7   (worth 0.25% added to the final grade)
Assigned on 04/   :           Due:  04/
Note: Before you attempt this extra-credit assignment, please read these instructions!
   a)  Use the fact that  A x* = y* ,  where  x*  is the LS solution of the  A x = y ,  to find an expression for  y* .
   b)  Compare it to the expression for the projection of  y  on a vector  A1  alone. That is, convince yourself
and then me that they can be written in the same form. (That should come as no surprise:  y*  is the projection
on the plane made by all columns of  A, so it should generalize the formula for a projection on one vector A1.)
   c)  Your expression for  y*  should look like  A B y , where  B  is called the pseudoinverse matrix of  A.
What is  B  when  A  is nonsingular?


HW # 17
Assigned on: 
Sec. 4.1:  ##   1, 3, 9, 13, 17, 18, 19.
                     Word problems
                     1.  Give a geometric proof that the matrix corresponding to the reflection about
                          any line passing through the origin in R^2 has an eigenvalue  lambda = -1.
                     2.  Is there such a rotation in R^2 that its matrix has  an eigenvalue  lambda = -1?
                          (Again, your answer should be based solely on geometric considerations.)
                          Answer:  Yes, by 180 degrees.
                     3.  Let  w  be any vector in  R^2  and  P  be the matrix of projection on w .
                          Give a geometric proof that this matrix has eigenvalues  0  and  1.


                     Note for # 17:   This asks you to show that the eigenvalues of A are always real.
                                            You'll need to use the identity  (a+d)^2 - 4ad = (a-d)^2.
                     Note for # 18:   This asks you to show that
the eigenvalues of A are always complex.
                    
Hint for all Word Problems:  
                                             Proceed as in Example 2 considered in class.
                    
Hint for Word Problem # 3:  
                                             You may benefit from reviewing a related Example found towards
                                             the end of posted Notes on Sec. 3.7.


HW # 18
Assigned on: 
Sec. 4.2:  ##   1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 24, 25, 26, 27, 29, 31.
                     Answer for # 2:  [-1 3 1; 2 4 1; 2 0 -2].
                     Answer for # 4:  [-1 3 1; 1 3 -1; 2 4 1].
                     Suggestion:  A useful Matlab command for this assignment is  det(A).
                     Hint for # 25:     Start with the definition of the inverse matrix
                                              found at the beginning of Sec. 1.9.
                     Note about # 31:   This exercise is important as it shows that a computation of
                                                 determinants of LARGE matrices in the way we learned here
                                                 is a VERY LONG process.  See the answers below.
                                                 Numerical software computes determinants using different techniques.
                     Answer for # 31:   (a)  n!/2;  
                                                (b)  n=2 => 3  ( = 2 multiplications + 1 addition),
                                                      n=3 => 3  * 3 determinants + 3 + (3-1)  = 9+3+2=14 s
                                                                 (the 3rd "3" and the following "3-1" occur because
                                                                  you need to multiply 3 determinants by a_{1j} and then
                                                                  add the 3 results);
                                                      n = 4 => 14 * 4 determinants + 4 + (4-1).
                                                      Continuing in this way, we see that the most time will be spent
                                                      on computing the  n!/2  determinants, and only a small portion of the
                                                      time will be spent on multiplying them by a_{1j} and adding.
                                                      Thus, to compute the determinant of a 10 X 10 matrix in this way,
                                                      one would need about  10!/2  > 1,800,000 s = 3 weeks!


HW # 19
Assigned on: 
Sec. 4.4:  ##   1, 3, 5, 7, 8, 11, 15, 17, 16, 18 [in regards to part (b), do it only for matrix A];
                     
p. 352 # 10;   p. 351 # 7 in Supplementary (not Conceptual!) Exercises.

                     Word problems:

                          1.   Let  v=[1  2]^T  and  lambda=3  be an eigenvector and the
                     corresponding eigenvalue of some  2 x 2  matrix  AIn addition, let it be known that
                     the number a = -4 is not an eigenvalue of A.
                     (a) Prove that C = (A+4*I) is nonsingular.
                     (b) Find a related eigenpair (i.e., eigenvector and eigenvalue related to v and lambda above)
                          of matrix 
                          B = C^(-1)   (i.e., of (A + 4*I)^(-1)).
                     (c) Find a related eigenpair of matrix
                          E = (A - I)^3 A^2 (A + I).

                         2.  Let A be an invertible  n x n  matrix with an eigenvalue Lambda.
                              Prove that  Lambda^k  is an eigenvalue of  A^k  for  k = -2, -3, -4, ...
                              (It will suffice for you to do the cases k=-2 and -3; then you will see the pattern.)

                          3.   Find all eigenvalues of each matrix (written in Matlab's notations):
                                (a) A = [1 -1 0;  1 3 0;  4 4 2],        (b) A = [2 0 2;  2 1 3;  5 0 -1].
                                Hint:   You must do an expansion of the determinant with respect to
                                           a specific row or column (different for (a) and (b)) to avoid
                                           factoring a cubic polynomial.

                     Note for ## 7, 8, 11, and Word Problem 3:
                                                        Decide which row or column is most convenient to use
                                                        for the co-factor expansion of the determinant.
                                                        (In some of these problems, there may be more than
                                                         one  convenient choice.)
                     Note for ## 7, 8:  In these problems you will end up factoring a cubic polynomial,
                                               but I expect that this will be easy to do. However, if you have
                                               difficulty factoring them, use Mathematica.
                     Note for # 11 and Word Problem 3:  
                                               In these problems you should not have to factor a cubic if you do
                                               these problems correctly, following the "Note for ## 7,8, 11, and WP3" above.
                     Answer for # 8:   lambda = 0, -1 (alg. multiplicity 2).
                     Reminder for all proofs:  Write down, in mathematical notations,
                                                          what you are given and what you want to prove.
                     Hint for # 16:   Use the distributive law to the expression on the left-hand side.
                     Clarification for # 17:   You need to give a proof for integer k<-1 (i.e., -2, -3, ...).
                                                       For that, you need to combine the ideas of proofs of
                                                       parts (i) and (ii) of Thm. 11, which were shown in class.
                     Hint for # 10 on p. 352: Use Theorem 11 to find an equation for the eigenvalue(s) of A.
                                                        For that, act with both sides of the given equation on an eigenvector
                                                        of  A,  similarly to what was done in Example 3 in the posted Notes.
                                                        (Also, note that you must not asume that A = O! You are not given
                                                         this information. All you are given is an equation from which you can
                                                         obtain an equation for the eigenvalues.
                                                         Counterexample to A = O: You may verify that A = [0, 1; 0, 0]
                                                         satisfies the given equation.)

                     Notes for #7 on p. 351: (a) The condition on matrix A is stated before # 6.
                                                        (b) You should not try to find A.
                     Hint
for #7  on p. 351:  If the Hint given in the book doesn't help, try the following.
                                                        (a) Use Theorem 11 to find eigenvalues of A;
                                                             follow the Hint for # 10 on p. 352.
                                                             In this case, the eigenvalues will be some irrational numbers,
                                                             which you should find using the quadratic formula.
                                                        (b) What eigenvalue(s) a nonsingular matrix cannot have?
                                                             (Equivalently, what eigenvalue(s) a singular matrix must have?
                                                              Find an answer in one of the Theorems in Sec. 4.4.)

                     Hints for Word Problem 1:  
                                                        (a) Use the definition of the eigenvalue and its relation with the
                                                              determinant. Then, use Theorem 3 and/or the definition of
                                                              eigenvalues via a determinant in Sec. 4.4. As an alternative, 
                                                              you may use Theorem 11 to make a conclusion about one of
                                                              the eigenvalues of C.  (Finally, if you are still stuck, looking at
                                                              proof of Theorem 13 in the Notes may be helpful, even though
                                                              you will not need the result that Theorem here.)

                                                        (b) Apply two parts of Thm. 11 one after the other.

                                                              Make sure that at each step, you act with your matrix on a certain eigenvector.
                                                        (c) Follow the method of Example 3 of Lecture Notes;
                                                              use Thm. 11 and the associative property of matrix multiplication as needed.
                     Answer for Word Problem 1(b):   v,  1/7.
                     Answer for Word Problem 1(c):   v,  288.
                     Hint for Word Problem 2:   Use Theorem 11(a) and then Theorem 11(b).
                     Answers for Word Problem 3:    (a)  lambda = 2 (alg. multiplicity 3);   (b)  lambda = -3, 1, 4.


Extra credit # 8   (each problem is worth 0.3% added to the final grade)
Assigned on:           Due: 
Note: Before you attempt this extra-credit assignment, please read these instructions!
Sec. 4.4:  ##   24,  27 (Hint for (c): proceed by induction.),  28.
                        Hints for # 24:   (a)  How are e1, e2, e3 related to I?
                                                    (b)  1. What does  p(lambda1)  equal to?
                                                              Same question for 
p(lambda2)  and  p(lambda3).
                                                           2. Compute  p(A)u1;  see a similar example done in class.
                                                               Repeat for
p(A)u2 and p(A)u3.
                                                           3.
In the book's hint, let "any vector" be e1
                                                               What is  p(A)e1 then? Why? Make sure you use the
                                                               book's hint in your explanation.
                                                               Continue in the same vein. 
                                                           4.  Now use the result of part (a) to prove Cayley--Hamilton for A.



HW # 20
Assigned on: 
Sec. 4.5:  ##   1, 3, 4, 6, 12, 13, 17, 19, 21, 22.
                     A useful Matlab command to check your answers is:  [V,L]=eig(A) .
                     Type  help eig  to learn the meaning of the output of this command.
                     Keep in mind that Matlab normalizes (scales) eigenvectors to have the length of 1.

                    
Reminder for all proofs:  Write down, in mathematical notations,
                                                          what you are given and what you want to prove.

                     Answer for # 4:   Algebraic multiplicity is 2, geometric multiplicity is 1.
                     Answer for # 6:   Algebraic multiplicity is 3, geometric multiplicity is 2.
                     Answer for # 12:   Not defective.
                     Discussion of # 21:   The name "idempotent matrix" is just a fancy name for
                                                    a projection matrix. Indeed, suppose we project a vector  x 
                                                    on some line  m  and obtain a vector  y  along line  m. Then
                                                    we can write:   Px=y.  But if now project  y  again on line  m,
                                                    we will still obtain  y.  Thus,  Py=y , or, using the definition of  y,
                                                    P2x == Px, which is the definition of an idempotent matrix.
                                                    We will examine projection matrices more closely in Sec. 4.7.
                      Additional hint for # 22:   Use the definition of an idempotent matrix and a result of Thm. 11.
                      Note for # 22:   The result you proved agrees with the fact that a projection matrix is singular
                                              (see class notes for Sec. 3.7), in view of Thm. 13.
            
                                     
HW # 21
PART 1

Assigned on:  
Sec. 4.7:  ##   1, 2, 3, 4, 5, 7, 9, 10, 11, 25, 26, 27.
                     General note for those problems out of 1--5, 7, 9--11 where A is diagonalizable:
                                         Contrary to what Ex. 1 in the book appears to suggest,
                                         you should NOT compute  D  as  V^-1 A V.
                                         This is because  D  is known to have a certain structure by its very design.
                                         In other words, if you have solved for eigenvalues and eigenvectors
                                         of A, you do NOT need to do any additional calculations to find  D.
                     Hints for # 10:  To determine whether A is diagonalizable, use the Matlab command
                                             eig, mentioned for Sec. 4.5  above.
                                             To check your answer for A^5, which you are to obtain by hand using
                                             the diagonalization of A, you may compute A^5 in Matlab.
                     Additional hint for # 26:   Substitute the expression for A into the expression for B.
                     Hint for # 27(a):   Proceed similarly to the proof of Thm. 18 (p. 326).

PART 2
Assigned on: 
Sec. 4.7:  ##   13, 14 [yes], 15, 17;  28, 29, 30;  33, 35, 36 [+ see Additional assignment below].
                     Additional assignment for ## 33, 35, 36
                                           Calculate A^5 by the same method as in ## 1--5 (see also Ex. 2 in the Notes).
                     see problems from Sec. 4.5 and from p. 352 listed below .

                     Hint for # 28:   Proceed as in the proof for part (a).
                     Hint for # 29:   Use the definition of an orthogonal matrix, i.e. compute  Q*Q^T.
                     Hint for # 30:   Use the definition of an orthogonal matrix, i.e. compute  (AB)*(AB)^T.
                     Notes for ## 33, 35, 36:  
                                            1.  Of course, you should follows the lines of an example in the Notes.
                                            2.  Unlike the General note for ## 1--5, 7, 9--11 above, here you
                                                 do have to compute  T.  Note, however, that you can still tell by
                                                 looking at your  T  whether it is likely correct or not.  (What should its
                                                 diagonal entries be?)
                     Answer for # 29:   Q u = - u; so 'yes', an eigenvector.
                     Answer for # 36:   There are several equivalent froms for the answer. I'll list two which are

                                                 lest similar to each other:
                                                 (1)  Q = [-1 1; 1 1]/Sqrt[2],    T = [0 1; 0 5].
                                                 (2)  Q = [2 -3; 3 2]/Sqrt[13],  T = [5 -1; 0 0].
                                                 You should be able to check your answer for A^5 using Matlab.

Sec. 4.5:  ##  23, 24, 25, 26, 27(b,c).
                    Note for ## 23, 25, 26: Do not assume a particular numeric form for u

                                                      (i.e., do not assume  u=[1; 0] or anything of that sort).

                                                      You will be requiredto carry out a proof with a general  u

                                                      satisfying the condition stated in the problem.

                                                      The method here is similar to that illustrated in Example 3 of Lec.~21.

                    Notes/Hints for # 24:    1. This problem actually has two parts, one part per sentence.
                                                      2.  For part (a), what does it mean that some matrix A is idempotent?
                                                      3.  For part (b): 
                                                           (i)Use the definition of the inverse matrix from the
                                                               posted Notes (not the textbook!) in Sec. 1.9. Namely, if B is
                                                               an inverse of A, what equation do A and B satisfy?
                                                          (ii) Now, if  C is the inverse of itself, what equation does C
                                                               satisfy?
                                                         (iii) Finally, what should the C be in this problem?
p. 352, # 14(a -- d).  Hint for (a):  Review Example 4 in the posted Notes for Sec. 1.6.
                               Answers:  (a) yes;  (b,c) see Answers to # 4.7.29;  (d) w .
                   


       Have  a  safe  and  relaxing break!