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 eve
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
S
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
(
Math 308 C - Summer 2016
Final Exam, Part 2 - August 19, 2016
Name:
Student ID no. :
Signature:
Section :
1
2
3
Total
20
15
15
50
This exam consists of 3 problems on 5 pages, including this cover she
Math 308 J, K - Spring 2017
Midterm 1 - April 19, 2017
Name:
Student ID no. :
Signature:
Section :
1
2
3
4
5
Total
10
8
6
12
13
49
This exam consists of 5 problems on 5 pages, including this cover sh
Math 308 J - Winter 2017
Midterm 1 - January 27, 2017
Name:
Student ID no. :
Signature:
Section :
1
2
3
4
5
Total
9
7
10
9
15
50
This exam consists of 5 problems on 6 pages, including this cover shee
Math 308 J - Winter 2017
Final Exam - March 14, 2017
Name:
Student ID no. :
Signature:
1
2
3
4
5
6
7
Total
12
12
14
15
8
12
27
100
This exam consists of 7 problems on 9 pages, including this cover sh
Math 308 C - Summer 2016
Final Exam, Part 1 - August 17, 2016
Name:
Student ID no. :
Signature:
Section :
1
2
3
Total
19
14
17
50
This exam consists of 3 problems on 7 pages, including this cover she
Name:
Math 308 J, K, Spring 2017
Proof HW #1
Due 4/14/15
Proof Homework 1
1. Let S = spancfw_u~1 , u~2 , . . . u~m .
(a) Show that ~0 S.
(b) Show that if ~v S, and c is a scalar, then c~v S.
(c) Show
Name:
Math 308 J,K, Spring 2017
Proof HW #2
Due 5/1/17
Proof Homework 2
WARNING: BE SURE TO USE ALL OF THE HYPOTHESES. If you do not, your proof is wrong.
Please try to recognize if your own attempt i
Math 308 J, K - Spring 2017
Midterm 2 - May 17, 2017
Name:
Student ID no. :
Signature:
Section :
1
2
3
4
5
Total
6
5
14
12
13
50
This exam consists of 5 problems on 5 pages, including this cover shee
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
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 cove
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
Math 308: Homework 3 Selected Solutions
Mary Radclie
a11 a12 a13
b11 b12 b13
1.5.66 Suppose A = 0 a22 a23 and B = 0 b22 b23 are 3 3 upper
0
0 a33
0
0 b33
triangular matrices. Then we have
a11 a12 a13
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
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
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 b
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 tran
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 for
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 3dimen
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 com
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,
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
Name:
Math 308 K, Autumn 2016
Proof HW #3
Due 11/21/16
Proof Homework 3 Solutions
1. Let cfw_a~1 , a~2 , . . . , a~n be a basis for Rn and let A be the matrix with these vectors as columns. Let
b~1 ,