Math 308: Homework 4 Selected Solutions
Mary Radclie
1.7.30. Recall that the vectors are linearly dependent if, when placed in the
columns of a matrix, the matrix does not contain a leading one in every
column after being reduced to echelon form. Here, we
Math 308C: Final Review
1. (a) Calculate the determinant of the following 4 4 matrix. You may use any techniques
developed in class to simplify your calculations.
1
2
4 1
3
6
2
1
A=
1
3 1 0
1 2 1 1
(b) Is 0 an eigenvalue of A? Why/why not?
2. For A as be
Math 308C: Final Review
1. (a) Calculate the determinant of the following 4 4 matrix. You may use any techniques
developed in class to simplify your calculations.
1
2
4 1
3
6
2
1
A=
1
3 1 0
1 2 1 1
Solution: We use row reduction to simplify the calculati
Math 308C: Midterm Review
The midterm on October 28 covers Sections 1.1-1.7, 1.9, 3.1-3.2. For topic-by-topic help,
please see this guide. The midterm will be about 50% computational exercises similar to
homework, and about 50% conceptual problems similar
This document is intended to help you focus your studies for the upcoming midterm. For
many of the fundamental course concepts, I have identied some helpful places to turn. This
contains material covered in Chapter 1 and Sections 3.1-3.2. Good luck!
So, y
Math 308C: Quiz 6
6 December 2013
5 3
3
3 .
1. Let A = 3 1
3
3 1
(a) (4 points) Find all the eigenvalues of A. (A hint to help you factor: 1 is one of
the eigenvalues.)
Solution:
5
3
3
1
3
0 = det(A I ) = det 3
3
3
1
5
3
3
1
3 by adding R2 to R3
= det
Name:
Student Number:
Math 308C: Quiz 2
7 October 2013
1
0
1. Let A =
0
0
01
1 2
00
00
0
0
1
0
02
0 3
.
0 1
14
(a) (4 points) If A is the augmented coecient matrix for a system of linear equations,
identify which variables are dependent and independent.
Name:
Student Number:
Math 308C: Quiz 3
14 October 2013
1. Suppose A, B, C are 2 2 matrices with products
AB =
1 1
,
34
BC =
32
,
1 1
and AC =
01
.
13
For each of the following expressions, either compute the expression or determine that
it is not possibl
Math 308C: Quiz 5
19 November 2013
0
4
1
0 , 1 , and v = 3. Determine the least-squares
1. (10 points) Let W = Sp
1
1
2
approximation to v in W by any method. (Note: your solution should be a 3dimensional vector in W ).
Solution: We rst orthogonalize th
Math 308C: Quiz 4 Solutions
6 November 2013
1. (8 points) Consider the matrix A =
1 2 1 1 2
. Find a basis for N (A) and
0 1 1 3 0
for R(A).
Solution: We rst consider the matrix in reduced echelon form:
1 2 1 1 2
0 1 1 3 0
10
1 5 2
0 1 1
30
This has a lea
Math 308: Homework 7 Selected Solutions
Mary Radclie
3.7.35. Let F : V W and G : V W be linear transformations, with F + G :
V W dened by [F + G](v) = F (v) + G(v). We show that F + G is a
linear transformation.
First, suppose v1 , v2 V . Then
[F + G](v1
Math 308: Homework 6 Selected Solutions
Mary Radclie
3.5.30 Let w1 W , with w1 = 0. Let S = cfw_w1 . If S is a spanning set for W ,
then S1 is a basis for W and we are done. If not, then S1 is not a basis
for W , so there exists w2 W such that w2 cannot b
Math 308: Homework 1 Selected Solutions
Mary Radclie
1.1.38. We consider two cases, according as whether a11 = 0 or a11 = 0.
First, if a11 = 0, we may form the augmented coecient matrix for the
system, and row reduce as follows:
a11
a21
a12
a22
b1
b2
R1
Math 308: Homework 2 Selected Solutions
Mary Radclie
1.3.8. The system has innitely many solutions. This is because there are more
variables than equations, so it cannot have exactly one solution, and as
it is homogeneous it cannot be inconsistent. Thus i
Math 308: Homework 5 Selected Solutions
Mary Radclie
3.3.51 Note that N (A) N (B ) = cfw_x R() n | x N (A) and x N (B ). Note
that if x N (A) N (B ), then (A + B )x = Ax + Bx = 0 + 0 = 0,
since x N (A) implies Ax = 0 and x N (B ) implies Bx = 0. Thus,
x N