Difference between revisions of "12-240/Classnotes for Tuesday October 30"

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{{12-240/Navigation}}
 
{{12-240/Navigation}}
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==Lecture Notes==
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Fix a linear transformation T:V->W
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'''Definition:'''
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N(T)=          ker T = {v ∈ V: Tv=O} ⊂ V
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"null space"  "kernal"
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R(T)=          img T = {Tv: v ∈ V} ⊂ W
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"range"        "image"
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'''Proposition/Definition'''
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1. N(T) is a subspace of V  nullity(T)= dim N(T)
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2. R(T) is a subspace of W    rank (T) = dim R(T)
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'''Example 1'''
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T= 0 of linear transformation  Tv=0
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ker T = N(T)= V  nullity(T) = dim V
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img T = R(T)={0}  rank (T)= dim{0}=0
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'''Example 2'''
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V=W; T=I Tv=V
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ker T = N(T)= {0} nullity(T) = 0
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img T = R(T)=  V  rank (T)= dim V
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'''Example 3'''
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V=Pn('''R''')= W; T=d/dx  T(x^3)=3(x^2)
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ker T = N(T)= {c(x^0): c∈'''R'''}  nullity(T) = 1
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img T = R(T)= Pn-1('''R''')          rank (T)  = n
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sum=n+1=dim V
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'''Theorem: Dimension Theorem/Rank-Nullity Theorem'''
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Given T:V->W, (V is finite dimensional)
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dim V = rank(T) + nullity (T)
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'''Corollary of Theorem'''
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If dim V = dim W then TFAE (the following are equivalent)
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1. T is 1-1
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2. T is onto
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3. rank (T) = dimV (maximal)
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4. T is invertible
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T is 1-1 <=> nullity (T) = 0 as n+r = dim V
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<=> rank(T) = dim V
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<=> T is onto
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1<=> 3
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invertible => 1-1 and onto
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1-1        => onto => invertible
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onto      => 1-1  => invertible
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== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==
 
== Lecture notes scanned by [[User:KJMorenz|KJMorenz]] ==
 
<gallery>
 
<gallery>

Latest revision as of 14:22, 8 November 2012

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 KJMorenz

Lecture notes uploaded by gracez