【英文原版书】Linear Algebra.pdf

Linear Algebra
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x
12

x
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62

81
Jim Hefferon
Notation
R real numbers
N natural numbers: {0, 1, 2,...}

plex numbers
{...
...} set of ... such that ...
h...i sequence; like a set but order matters
V,W,U vector spaces
~v,w ~ vectors
~0, ~0V zero vector, zero vector of V
B,D bases
n
En = h~e1, ... ,~eni standard basis for R

,~ ~
basis vectors
RepB(~v) matrix representing the vector
Pn set of n-th degree polynomials
Mn
m set of n
m matrices
[S] span of the set S
M
N direct sum of subspaces

V = W isomorphic spaces
h, g homomorphisms
H, G matrices
t, s transformations; maps from a space to itself
T,S square matrices
RepB,D(h) matrix representing the map h
hi,j matrix entry from row i, column j
|T | determinant of the matrix T
R(h), N (h) rangespace and nullspace of the map h
R∞(h), N∞(h) generalized rangespace and nullspace
Lower case Greek alphabet
name symbol name symbol name symbol
alpha
iota
rho
beta
kappa
sigma
gamma
lambda
tau
delta
mu
upsilon
epsilon
nu
phi
zeta
xi
chi
eta
omicron o psi
theta
pi
omega ω
Cover. This is Cramer’s Rule applied to the system x +2y =6,3x + y =
of the
rst box is the determinant shown. The area of the second box is x times that,
and equals the area of the
nal box. Hence, x is the
nal determinant divided by the
rst determinant.
Preface
In most mathematics programs linear algebra is taken in the
rst 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 in-
structors have experimented with varying the traditional topics, trying courses
focused on applications, or on puter. Despite this (entirely healthy)
debate, most instructors are still convinced, I think, that the right core material
is vector spaces, linear maps, d