Introduction

Linear algebra is the subject that deals with linearity. This means lines, planes and other flat things in higher dimensions, and functions sending flat things to flat things. The key foundational concepts here are that of the vector space and that of the linear map. Most introductory linear algebra books treat the subject as purely computational. Finding way to compute things and leave it at that. For the mathematically inclined this is insufficient, as it doesn’t show why it is true nor does it give deeper insight. This is why an axiomatic approach using vector spaces right away is preferable.

Problem Selection

Note: this is for fourth edition.

Chapter 1: Vector Spaces

Section 1.1: Introduction

This section is not essential for the rest of the book and is purely motivational. It can safely be skipped for those familiar to vectors in say physics or even high school.
1ab C
2a C
3a C

Section 1.2: Vector Spaces

1 F
2 C 3 C 4ace C
5 A
7 C
8 F
9 F
10 M
11 F
13 F
14 F
15 F
20 M
21 F
22 M

Section 1.3: Subspaces

1 F
2abde C
3 F
4 F
5 F
6 F
7 F
8abc C
10 F
11 F
13 M
14 M
16 M
17 F
18 F
19 F
20 F
21 M
23 MP
24 MP
28 MP
30 MP
31 M

Section 1.4: Linear Combinations and Systems of Linear Equations

1 F
2ab C
3ab C
4ab C
5aeg C
6 C
8 F
9 F
10 F
11 F
13 F
14 F
15 F

Section 1.5: Linear Dependence and Linear Independence

1 F
2acegi C
3 C
4 F
6 F
7 F
8 M
9 F
10 F
11 M
12 F
13 M
15 F
16 F 18 M
19 F
20 C

Section 1.6: Bases and Dimension

1 F
2ab C
3ab C
4 C
5 C
7 C
9 C
11 F
13 C
15 F
19 F
20 F
21 F
25 M
26 M
29 MP
31 MP
33 MP
34 MP
35 M

Section 1.7: Maximal Linearly Independent Subsets

I suggest skipping this section for anybody not in the M track.
1 M
2 M
3 M
4 M
5 M
6 M
7 M

Chapter 2: Linear Transformations and Matrices

Section 2.1: Linear Transformations, Null Spaces, and Ranges

1 F
2 C
3 C
4 C
5 C
6 F
7 F
9 F
10 C
13 F
14 F
15 MP
16 MP
17 F
20 F
21 MP
22 F
25 MP
36 MP
28 M
29 M
32 M
34 F
40 M

Section 2.2: The Matrix Representation of a Linear Transformation

1 F
2abg C
5 C
6 F
7 F
8 F
12 MP

Section 2.3: Composition of Linear Transformations and Matrix Multiplication

1 F
2 C
4ab C
5 F
6 F
7 F
12 F
13 F
18 F
19 A
20 A

Section 2.4: Invertibility and Isomorphisms

1 F
2adef C
3 C
4 F
5 F
6 F
7 F
8 F
12 F
14 C
15 F
16 F
17 F
19 C
20 F
22 M
24 M
25 M

Section 2.5: The Change of Coordinate Matrix

1 F
2ab C
3ab C
6ab C
7ab C
8 F
9 F
10 F
12 F
14 F

Section 2.6: Dual Spaces

This should be skipped by anybody except the M and P tracks.
1 MP
2ab MPC
3 MPC
4 MPC
5 MP
10 M
11 M
14 M
15 M
16 M

Section 2.7: Homogeneous Linear Differential Equations with Constant Coefficients

Even though Friedberg considers this section optional, I do recommend it for everybody
1 F
3abc C
4a C
5 F
6 F
8 F
10 F
11 F
13 F
14 F
14 F
16 P
17 P
18 P