Math 441/541 Assignment 1: Solutions
Chapter 1 3. Here we use the associativity of scalar multiplication and the fact that 1 v = v . (v ) = 1 (v ) = 1 (1 v ) = (1 1)v = 1 v = v.
4. Let a F and v V and
Math 307 Homework
November 4, 2015
1. Show that A Mn (C) is unitary if and only if j = 1 for j = 1, . . . , n.
2. Show that if A = diag(1 , . . . , n ), then the singular values of A are |1 | , . . .
Math 307 Homework
August 30, 2015
1. Prove that any two non-zero vectors in R2 which are not collinear span R2 .
Hint: The hard part here is figuring out how to express and then use the fact
that the
Math 307 Homework
August 18, 2015
1. Give examples of linear systems of each of the following types, if possible. Explain how you know they have the claimed properties, or else explain why there
is no
Math 307 Homework
August 26, 2015
1. Suppose that f (x) = ax2 + bx + c is a quadratic polynomial whose graph passes
through the points (1, 1), (0, 0), and (1, 2). Use this information to write down
a
Group
Names
Group Quiz for Section 1.6
Let C(R) be the vector space of all continuous functions f : R R. For f C(R),
dene T f : R R by (T f )(x) = f (x) cos x.
Show that T is a linear map from C(R) to
Math 307 Homework
August 24, 2015
1. Suppose that x1 = c1 , . . . , xn = cn is a solution of the linear system
a11 x1 + + a1n xn = b1 ,
.
.
.
am1 x1 + + amn xn = bm .
Under what circumstances is x1 =
Names
Group Quiz for Section 1.4
We use F3 to stand for the set cfw_0, 1, 2 with the following addition and multiplication tables:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
0
1
2
0
0
0
0
1
0
1
2
2
0
2
1
Which e
Practice problems for exam 1
1. Is the assignment f : R, R2
f (x, y) = z,
|z| = x2 y 2
a function? Justify your answer.
No, because z = x2 y 2 , so to one vector (x, y) we could associate two differen
Math 307 Homework
September 25, 2015
1. Two nonzero vectors v, w V are called collinear if there is a scalar c F
such that v = cw.
(a) Given two nonzero vectors v, w V , prove that (v, w) is linearly
Math 307 Homework
October 12, 2015
1. Let B = 1, x, x2 and let B0 = 1, x, 32 x2 12 in P2 (R). Find the change of
basis matrices [I]B,B0 and [I]B0 ,B .
1 1
2. Let A =
.
0 1
(a) Find all the eigenvalues
Group
Names
Group Quiz for Section 1.4
We use F3 to stand for the set cfw_0, 1, 2 with the following addition and multiplication tables:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
0
1
2
0
0
0
0
1
0
1
2
2
0
2
1
W
Group
Names
Group Quiz for Section 1.3
1
1
3 and 0 in R3 ? Justify your answer.
What is the span of
0
0
Note: to prove two sets are equal, you have to prove they are exactly the same; that
is, tha
Group
Names
Group Quiz for Section 1.5
Let A Mn (F) (that is, A is an nn matrix over F). The trace of A = [aij ]1i,jn
is dened to be
n
tr(A) :=
aii .
i=1
Show that the set
cfw_A Mn (F) : tr(A) = 0
is
Group
Names
Group Quiz for Section 1.2
Determine whether each of the matrices below is in row-echelon form, reduced
row-echelon form, or neither. For any matrices in RREF, solve the corresponding
syst
Math 307 Homework
September 9, 2015
1. Prove part 1. of Proposition 1.13.
2. Suppose that T L(U, V ) and S L(V, W ).
(a) Show that if ST is injective, then T is injective.
(b) Show that if ST is surje
Math 307 Homework
November 20, 2015
1. Let A 2 Mn (C) and let " > 0. Show that there is a B 2 Mn (C) with n distinct
eigenvalues such that kA Bk ".
Hint: First consider the case where A is upper trian
Math 307 Homework
November 6, 2015
1. Prove that if A Mn (C) has singular values 1 , . . . , n , then
|tr A|
n
X
j .
j=1
Hint: Use SVD (in the form of Corollary 3.31) and the CauchySchwarz inequality
Math 307 Homework
October 14, 2015
1. Give an example of matrices with the same trace which are not similar.
2. Give an example of three square matrices A, B, and C of the same size, such
that tr ABC
Math 307 Homework
September 2, 2015
1. Determine which of the following are and are not subspaces of the given vector
space, and justify your answers.
(a) The x axis in R3 .
x
(b) The set
x, y 0 in
Checklist for Midterm 1
In preparation for the first exam, here is a list of the material that we have covered in class and that you
need to know well. I am working at a practice exam or two that you
Practice problems for exam 1
1. Is the assignment f : R, R2
f (x, y) = z,
|z| = x2 y 2
a function? Justify your answer.
2. Show that the composition of function is not commutative by finding two funct
Math 307 Fall 2014 Exam 1 Practice
Definitions provided:
Definition of Field. A field consists of a set F and two binary operations + (addition) and (multiplication), for which
the following conditio
MATH 307 FALL 2014
Homework 14
1. Find a SVD of matrix
[
]
.5 1
A=
,
0.5 1
1 2 3
2. A singular value decomposition of matrix A = 2 3 8 is U V T
5 1 3
0.2871 0.4773 0.8305
10.0571
0
0
where U = 0.8640
MATH 307 FALL 2014
Homework 12
1. Tuesday material: A square matrix A Rnn is called orthogonal if A A =
I (thus AA = I). The analogue of orthogonal is unitary for complex matrices,
i.e., a square matr
MATH 307 FALL 2014
Homework 13
1. Let A be a symmetric real matrix. Prove that eigenvectors corresponding to
two different eigenvalues of A are orthogonal, i.e., let x1 , x2 be eigenvectors
correspond
MATH 307 FALL 2014
Homework 15
1. Tuesday material: Use SVD to solve Ax = b with matrix
[
]
[
]
.5 1
1.5
A=
,b =
.
0.5 1
0.5
You may use the SVD you found in HW14 without recomputing it.
2. Tuesday ma
Math 307 Fall 2014 Exam 2 Practice (Covers Chapter 3)
You are encouraged to review notes and homework problems thoroughly before working on practice problems.
This list of practice problems is just to
Math 307 Fall 2014 Final practice. This list only covers topics after exam 3.
Refer to practice for exam 1, 2, and 3 for other topics.
You are encouraged to review notes and homework problems thorough