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: 09/02
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 before # 30.
HW
# 2
Assigned on: 09/04
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: 09/04, Due:
09/11
Before you attempt this
extra-credit assignment, please read these
instructions!
Sec. 1.2: ## 49 (0.1%),
51 (0.15%), 56 (0.15%).
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 do not use
the REF.) If you
used Matlab for that, you need to attach a printout.
Brief solutions, found in
the Student's Manual, will receive no
credit.
HW
# 3
Assigned on:
09/09
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: 09/14
Sec. 1.5: ## 5, 9, 11, 21, 33,
35; 29, 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(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: 09/16
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.
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
general 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.
Hint for # 41(a): Write this as a linear system.
Hint for # 43(b): 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: 09/16
Due:
09/23
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: 09/23
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 # 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:
09/23
Due:
09/30
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: 09/25
Sec. 1.8: ## 1, 5, 27, 7.
HW
# 8
Assigned on: 09/30
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.
Note for ## 35, 37, 38, 39, 41, 43, 45:
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.
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: 09/30
Due: 10/07
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: 10/02
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: 10/07
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).
Hint for # 3 and both Word
Problems:
See one of the required
examples in Sec. 3.2 of the textbook (you need to find it).
Begin by plotting
W, following the description in that Example. Then try to give
a geometric proof. (A similar proof was done in an
example in the Notes towards
the end of 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: 10/09
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: {x: x=[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: 10/14
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:
If
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 class. Thus,
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.
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.
Hint: In
all three problems, proceed as in Example 2
considered in
class.
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.
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],
20, 24.
Word problems:
1.
Let v=[1
2]^T and lambda=3 be an
eigenvector and the
corresponding eigenvalue of some
2 x 2 matrix A. In 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) of matrix
B = C^(-1) (i.e., of (A + 4*I)^(-1)).
2.
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 2:
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 2:
In these problems you should not have to factor a cubic if you
do these problems correctly, following the Note 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.
Additional 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.
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.
Answer
for Word Problem 1(b):
v, 1/7.
Answers for Word Problem
2: (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: ## 27 (Hint for (c): proceed by
induction.), 28.
HW
# 20
Assigned on:
Sec. 4.5: ## 1, 3,
4, 6, 12, 13, 17, 19,
21, 22, 23, 24.
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.
Hint for # 24: Use the definition of the inverse
matrix
(see Sec. 1.9).
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 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 # 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, 25, 26,
27(b,c).
Have
a safe and relaxing break!