# 12-240/Classnotes for Tuesday October 30

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## Lecture Notes

Fix a linear transformation T:V->W

Definition:

N(T)= ker T = {v ∈ V: Tv=O} ⊂ V "null space" "kernal"

R(T)= img T = {Tv: v ∈ V} ⊂ W "range" "image"

Proposition/Definition

1. N(T) is a subspace of V nullity(T)= dim N(T) 2. R(T) is a subspace of W rank (T) = dim R(T)

Example 1

T= 0 of linear transformation Tv=0

ker T = N(T)= V nullity(T) = dim V

img T = R(T)={0} rank (T)= dim{0}=0

Example 2

V=W; T=I Tv=V

ker T = N(T)= {0} nullity(T) = 0

img T = R(T)= V rank (T)= dim V

Example 3

V=Pn(R)= W; T=d/dx T(x^3)=3(x^2)

ker T = N(T)= {c(x^0): c∈R} nullity(T) = 1

img T = R(T)= Pn-1(R) rank (T) = n

sum=n+1=dim V

Theorem: Dimension Theorem/Rank-Nullity Theorem

Given T:V->W, (V is finite dimensional)

dim V = rank(T) + nullity (T)

Corollary of Theorem

If dim V = dim W then TFAE (the following are equivalent)

1. T is 1-1

2. T is onto

3. rank (T) = dimV (maximal)

4. T is invertible

T is 1-1 <=> nullity (T) = 0 as n+r = dim V

<=> rank(T) = dim V

<=> T is onto

1<=> 3

invertible => 1-1 and onto

1-1 => onto => invertible

onto => 1-1 => invertible