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== | ||
+ | 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]] == | == Lecture notes scanned by [[User:KJMorenz|KJMorenz]] == | ||
<gallery> | <gallery> |
Latest revision as of 15:22, 8 November 2012
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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