MATH 225 EXAM 2 NAME 69mm a: A
Show all relevant work as neatly and completely as possible, and indicate your nal
answers clearly. Good luck.
1 2 0
1. [ZOpoints] Let A: 2 4 1 10 3 .
3 6 1
a) Find a basis for the row space of A.
. , l '2 C) 3 \
Q (33 Q :5)

MATH 225
Homework 2
1. The trace of a square matrix A is the sum of the diagonal entries. That is, if
then
a)
tr ( A) aii a11 a 22 a nn
.
A [ aij ]
is n x n,
Prove the following statements:
tr ( A B) tr ( A) tr ( B )
b) For any scalar k,
tr ( kA) k tr ( A

MATH 225
1. Let
3 2 6 1 15
A 4 3 8 10 14
2 3 4 4 20
, and define
Homework 10
T (x) Ax.
a)
Find ker(T).
b) Find the nullity of T.
c) Find the range of T.
d) Find the rank of T.
2. Given that
3. Let
T : P3 P1
1 2 3
A 1 2 4
0 4 1
is linear and rank(T)

MATH 225
EXAM 1
NAME _
Read each question carefully before answering. Show all relevant work as neatly and
completely as possible, and indicate your final answers clearly. Good luck.
x1
3 x3 2
1. [10 points] Given the system: 3 x1 x 2 2 x3 5 .
2 x1 2 x 2

MATH 225
Homework 11
1. A square matrix is A is called idempotent if A2 = A. Prove that if A is idempotent, and
B is similar to A, then B is also idempotent.
2. Prove that if A is similar to B, then Ak is similar to B.
3. TRUE or FALSE: For each of the fo

MATH 225
Homework #1
Due Tuesday Jan. 26
1. Solve the system of equations using either Gaussian or Gauss-Jordan elimination.
Show all steps.
2 x 3 y z 10
2 x 3 y 3 z 22
4 x 2 y 3 z 2
2. Solve the system of equations using either Gaussian or Gauss-Jordan e

MATH 225
EXAM 2
NAME _
Show all relevant work as neatly and completely as possible, and indicate your final
answers clearly. Good luck.
1
2
A
1. [20 points] Let
3
2
4
0
1
6
1
5 1
10 3 .
15 4
a) Find a basis for the row space of A.
b) Find a basis for the

Syllabus for Math 225
College of Lake County
Spring 2016
Tuesday/Thursday 12:30-1:45 pm A124
PROFESSOR: Mark Beintema
OFFICE:
A136
PHONE:
(847) 543-2913
e-mail:
mbeintema@clcillinois.edu
OFFICE HOURS: Monday, Wednesday: 9:30 am noon
Tuesday, Thursday 10 a

MATH 225
Homework 9
1. Let V be an inner product space. Prove that if w is orthogonal to each vector in
S = cfw_v1, v2, . . . , vn, then w is also orthogonal to any linear combination of vectors
from S.
2. Let P be an n x n matrix. Prove that the followin

MATH 225
Homework 3
1. Find the inverse of the elementary matrix:
a)
1
0
E
0
0
b)
0 3 0
1 0 0
0 1 0
0 0 1
1
0
E
0
0
0 0
1 0
0 1/ k
0
0
0
0
0
1
2. Prove that if A is row-equivalent to B, then B is also row-equivalent to A.
3. A researcher is studying the