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 suppose av = 0. If a = 0 then were done. So assume 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 C(R).
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 linear system satised by (a, b, c), and nd a solution f
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 such system.
(a) Underdetermined and inconsistent.
(b)
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 vectors are not collinear.
2. Let F = cfw_a + b 5 : a,
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 | , . . . , |n |
(though not necessarily in the same order).
3. P
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 dependent
if and only if v and w are collinear.
(b) Giv
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 and eigenvectors of A.
Hint:
You can
just work direc
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 R2 (i.e., the first quadrant of the plane).
y
x
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 6= tr ACB.
3. For A Mn (F), define
sum A =
n X
n
X
i=1
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.
2. Suppose that V is a complex inner product space, T
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 triangular, then use the Schur
decomposition.
2. (a) Prove t
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 surjective, then S is surjective.
3. Suppose that T L(V ) 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
system or explain why the system is inconsistent.
1 0 0
1 3
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 a subspace of Mn (F).
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, that each is a subset of the other.
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 = 2c1 , . . . , xn = 2cn also a solution?
2. Give a geome
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 element (0, 1, or 2) is each of the following equal to (
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Erkki Somersalo
November 30, 2016
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Contents
ENGR 145 Chemistry of Materials
Spring 2017 (rev. 17 Jan. 2017)
ENGR 145 Chemistry of Materials
Lectures M,W,F 10:3511:25, Strosacker Auditorium Recitations and tests as listed below
Lecturer:
Professor Mark De Guire
White 510
216-368-4221
[email protected]
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 thoroughly before working on practice problems.
Formula for ort
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 give you an idea on the format of the actual exam. It
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 material: Assume A Rmn , m n is of full rank, use reduced
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
corresponding to eigenvalues 1 , 2 , then x1 and x2 are orthogona
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 matrix A Cnn is called unitary if A A = I (thus AA = I).
Pr
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 0.2453 0.4397 , =
0
4.9855 0 ,
0.4136 0.8438 0.3420
0
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 conditions are satisfied:
1. Additive associative: (a + b) + c
MATH 307 FALL 2014
Homework 1
1. Prove that the set of complex numbers with the following regular multiplication and addition is a field:
(a + bi) + (c + di) = (a + c) + (b + d)i,
(a + bi)(c + di) = (ac bd) + (bc + ad)i
.
2. Let z = 1 + 3i, w = 3 i, find
A Singularly Valuable Decomposition: The SVD of a Matrix
Dan Kalman
The American University
Washington, DC 20016
February 13, 2002
Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or
SVD). It has interesting
Practice problems for Final Exam
1. If A is an m n matrix, define its rank and show that the rank of A is equal to the rank of AT A.
2. If A = QR is the QR factorization of the real m n matrix A, show that the least squares solution of
Ax = b exists and i
GramSchmidt Process and QR Decomposition
Recall from class that the GramSchmidt process takes a basis cfw_61, . . . ,mp for a sub
space W of R and produces an orthogonal basis cfw_01, . . . ,vp for the same subspace W,
where
01 = 331
$2 711
02 = $
Math 307 Linear algebra
Fall 2016
Practice problems
1. In R4 is the subset of vectors such that
x3 x4 = 1
a subspace? Justify your answer.
2. In R3 , is the subset of vectors such that
x2 = 3x1
a subspace? Justify your answer.
3. In R3 is the subset of ve