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
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