Axler Linear Algebra Chapter 3

Linear Map

Function from V to W with additivity and homogeneity

L(V, W) is a vector space by...

scalar multiplication and addition

Product of Linear Maps

If T in L(U, V), and S in L(V, W), then ST in L(U, w) is (ST)(u) = S(Tu)

Properties of Linear Map Products

Associativity

Identity

Distributive

Linear Maps map 0 to 0

T(0) = 0

Null space

Subset of V that T maps to 0

The null space is a subspace

Injective

A function T:V ->W such that Tu = Tv implies u = v

(aka null space = 0)

Range

A function T:V -> W, subset W consisting of vectors that are of form Tv for some v in V

is a subspace of W

Surjective

A function T:V -> W if range = W

Fundamental theorem of linear maps

dim V = dim(nullT) + dim(rangeT)

A map to a smaller dimensional space is not...

Injective

A map to a larger dimensional space is not ...

Surjective

Homogenous System properties

More variables than equations: nonzero solutions

Inhomogenous system properties

More equations than variables:

No solution for some choice of constant terms

Inverse

A linear map satisfying ST = I and TS = I

An invertible map has a...

unique inverse

Invertibility is equivalent to....

Invertibility and surjectivity

Isomorphism

An invertible linear map

Finite-dimensional vector spaces over F are isomorphic iff...

They have the same dimension

L(V, W) and F^{m, n} are...

Isomorphic

dim(L(V, W)) =

(dimV)(dimW)

Operator

Linear map from a vector space to itself:

L(V) = L(V, V)

INvertibility is equivalent to:

Injectivity and surjectivity