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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



== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==
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== Lecture notes uploaded by [[User:Grace.zhu|gracez]] ==
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