Math 223 Homework 6 Solutions
Problem 1: Let U and V be vector spaces and let F : U V be a linear transformation.
Show that the kernel of F is a subspace of U .
Solution 1: We have to show that ker(F ) is closed under addition and scalar multiplication.
T

Midterm Vocabulary List
October 14, 2013
Introduction
The following vocabulary list is designed to help you review for the midterm. With each
of the following terms, check that you understand what the term means and how to use it
in a mathematical sentenc

Math 223 Homework 10
Due in class Nov. 21
Problem 1 Let P2 (t) denote the space polynomials of degree at most 2, and let T : P2 (t)
P2 (t) be given by
t
df
T (f (t) =
1
(t)
2
dt
Does there exist f P2 (t) such that T (f ) = f ? If so nd such an f .
Does

Math 223 Homework 11
Due in class Nov. 28
Problem 1: Let A be an n n matrix, and let v be an eigenvector of A with eigenvalue
. Prove that if f (t) is any polynomial, then f (A)v = f ()v .
Solution 1: Note that since v is an eigenvector with eigenvalue ,

Midterm Vocabulary List
December 4, 2013
Introduction
The following vocabulary list is designed to help you review for the nal exam.
The sections of the book that we have covered in class since the midterm are the following:
Chapter 7 sections 1-9.
Chap

Math 223 Homework 9
Due in class Nov. 14
Recall that for an n n matrix A = [aij ], the determinant of A satises the following
equality:
n
(1)i+1 ai1 det(Mi1 ),
det(A) =
(1)
i=1
where Mi1 denotes the i, j -minor of M .
Problem 1: Let A = [aij ] be a 3 3 ma

Math 223 Homework 8
Due in class Nov. 7
Problem 1: Let V = P3 (t), the space of degree 3 polynomials with real coecients, and
let , be the inner product dened by
1
f, g =
f (t)g (t)dt
0
Apply the Gram-Schmidt process to cfw_1, t, t2 , t3 to obtain an ort

Math 223 Homework 2
Due in class Sept. 19
Problem 1: Let
2 3 1i
ii
0 , B = i 0 , C =
A= 1 0
0i
0
0i
111
i 2i i
Compute each product in the follow list that is well-dened:
ABC
BAC
CBA
BCA
ACB
CAB
Solution 1:
ABC , BCA and CAB are well-dened, and
3 +

Math 223 Homework 3
Due in class Sept. 26
The goal of the next 6 problems is to give a proof of the following theorem, which came up
in class:
Theorem: Let A be an n n matrix that is not invertible. Then there exists a vector v
with v = 0 such that Av = 0

Math 223 Homework 4
Due in class Oct. 3
In Problem 1, let P2 (t) denote the space of degree 2 polynomials with real coecients.
Problem 1: Let f (t) = 2t2 + t, g (t) = t2 t + 1, and h(t) = t 3. Do f, g and h span
P2 (t)? If so, write t2 + t + 1 as a linear

Math 223 Homework 5 Solutions
Problem 1: Let V be a vector space and let cfw_v1 , . . . , vk be a linearly independent subset
of V . Suppose that w V is not in the span of cfw_v1 , . . . , vk . Show that cfw_v1 , . . . , vk , w is
linearly independent.
S

Math 223 Homework 7
Due in class Oct. 31
Problem 1: Let
and
1
2
1
S1 = 2 , 1 , 2
0
1
1
1
1
0
S2 = 1 , 0 , 1
0
1
1
(a) Show that S1 and S2 are bases for R3 .
(b) Let Id : R3 R3 denote the identity map, i.e. the map dened by Id(v ) = v for all
v R3 .