Linear algebra

Linear Algebra

12
31

x·1 2 x·3 1

6
8

Jim Hefferon

2
1

Notation
R
N
C
{. . . . . . }
...
V, W, U v, w
0, 0V
B, D
En = e1 , . . . , en β, δ
RepB (v )
Pn
Mn×m
[S ]
M ⊕N
V ∼W
=
h, g
H, G t, s
T, S
RepB,D (h) hi,j |T |
R (h), N (h)
R∞ (h), N∞ (h)

real numbers natural numbers: {0, 1, 2, . . . } complex numbers set of . . . such that . . . sequence; like a set but order matters vector spaces vectors zero vector, zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree polynomials set of n × m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms matrices transformations; maps from a space to itself square matrices matrix representing the map h matrix entry from row i, column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace

Lower case Greek alphabet name alpha beta gamma delta epsilon zeta eta theta symbol α β γ δ ζ η θ name iota kappa lambda mu nu xi omicron pi

symbol ι κ λ µ ν ξ o π

name rho sigma tau upsilon phi chi psi omega

symbol ρ σ τ υ φ χ ψ ω

Cover. This is Cramer’s Rule applied to the system x + 2y = 6, 3x + y = 8. The area of the first box is the determinant shown. The area of the second box is x times that, and equals the area of the final box. Hence, x is the final determinant divided by the first determinant.

Preface
In most mathematics programs linear algebra is taken in the first or second year, following or along with at least one course in calculus. While the location of this course is stable, lately the content has been under discussion. Some instructors have experimented with varying the traditional topics, trying courses focused on applications, or on the computer. Despite this (entirely healthy) debate, most instructors are still convinced, I think, that the right