# Linear algebra

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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 maph
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
chipsi
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 ﬁrstor 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 theright core material
is vector spaces, linear maps, determinants, and eigenvalues and eigenvectors.
Applications and computations certainly can have a part to play but most mathematicians agree that the themes of the course should remain unchanged.
Not that all is ﬁne with the traditional course. Most of us do think that
the standard text type for this course needs to be reexamined. Elementarytexts have traditionally started with extensive computations of linear reduction,
matrix multiplication, and determinants. These take up half of the course.
Finally, when vector spaces and linear maps appear, and deﬁnitions and proofs
start, the nature of the course takes a sudden turn. In the past, the computation
drill was there because, as future practitioners, students needed to be fastand
accurate with these. But that has changed. Being a whiz at 5 × 5 determinants
just isn’t important anymore. Instead, the availability of computers gives us an
opportunity to move toward a focus on concepts.
This is an opportunity that we should seize. The courses at the start of
most mathematics programs work at having students correctly apply formulas
and algorithms, and imitateexamples. Later courses require some mathematical
maturity: reasoning skills that are developed enough to follow diﬀerent types
of proofs, a familiarity with the themes that underly many mathematical investigations like elementary set and function facts, and an ability to do some
independent reading and thinking, Where do we work on the transition?
Linear algebra is an ideal spot. It comes early ina program so that progress
made here pays oﬀ later. The material is straightforward, elegant, and accessible. The students are serious about mathematics, often majors and minors.
There are a variety of argument styles—proofs by contradiction, if and only if
statements, and proofs by induction, for instance—and examples are plentiful.
The goal of this text is, along with the development of...