Chapter 2.1 Definitions (Linear Transformations, Null Space, and Ranges)

Linear Transformation

Let V and W be vector spaces over the same field F. We call a function T: V → W a _____ ________ from V to W if, for all x,y∈V and c∈F, we have

a) T(x+y)=T(x)+T(y)

b) T(cx)=cT(x)

Linear

T is _____ if all the following properties hold

1) T(0)=0

2) T(cx+y)=cT(x)+T(y) for all x,y∈V and c∈F

3) T(x-y)=T(x)-T(y)

4) T(∑aixi)=∑aiT(xi)

Null Space/Kernel

Defined as N(T) of T. It’s the set of all vectors x∈V such that T(x)=0; that is, N(T)={x∈V: T(x)=0}

Range/Image

The ____ R(T) of T is the subset of W consisting of all images (under T) of vectors in V; that is, R(T)={T(x): x∈V}

Theorem 2.1

Let V and W be vector spaces and T: V → W be linear. Then N(T) and R(T) are subspaces of V and W, respectively

Theorem 2.2

Let V and W be vector spaces, and let T: V → W be linear. If B={v1, …, vn} is a basis for V, then

R(T)=span(T(B))=span({T(v1), …, T(vn)})

Nullity, Rank

_____ of T, denoted nullity(T), and the rank of T, denoted rank(T), are the dimensions of N(T) and R(T), respectively.

Theorem 2.3 (Dimension Theorem)

Let V and W be vector spaces, and let T: V → W be linear. If V is finite-dimensional, then

nullity(T)+rank(T)=dimV

Theorem 2.4

Let V and W be vector spaces, and let T: V → W be linear. Then T is one-to-one iff N(T)={0}

Theorem 2.5

Let V and W be finite dimensional vector spaces of equal dimension, and let T: V → W be linear. Then the following are equivalent

a) T is one-to-one

b) T is onto

c) rank(T)=dimV

Theorem 2.6

Let V, W be vector spaces over F, and suppose that {v1, …, vn} is a basis for V. For w1,…,wn∈W, there exists exactly one linear transformation T: V → W such that T(vi)=wi, for i=1,2,…,n

Corollary: If U,T: V → W are linear and U(vi)=T(vi) for i=1,2,…,n then U=T