Difference between revisions of "12240/Classnotes for Tuesday October 30"
Line 1:  Line 1:  
{{12240/Navigation}}  {{12240/Navigation}}  
+  
+  ==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)= Pn1('''R''') rank (T) = n  
+  
+  sum=n+1=dim V  
+  
+  '''Theorem: Dimension Theorem/RankNullity 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 11  
+  
+  2. T is onto  
+  
+  3. rank (T) = dimV (maximal)  
+  
+  4. T is invertible  
+  
+  
+  T is 11 <=> nullity (T) = 0 as n+r = dim V  
+  
+  <=> rank(T) = dim V  
+  
+  <=> T is onto  
+  
+  1<=> 3  
+  
+  
+  
+  invertible => 11 and onto  
+  
+  11 => onto => invertible  
+  
+  onto => 11 => invertible  
+  
+  
+  
== Lecture notes scanned by [[User:KJMorenzKJMorenz]] ==  == Lecture notes scanned by [[User:KJMorenzKJMorenz]] ==  
<gallery>  <gallery> 
Latest revision as of 15:22, 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)= Pn1(R) rank (T) = n
sum=n+1=dim V
Theorem: Dimension Theorem/RankNullity 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 11
2. T is onto
3. rank (T) = dimV (maximal)
4. T is invertible
T is 11 <=> nullity (T) = 0 as n+r = dim V
<=> rank(T) = dim V
<=> T is onto
1<=> 3
invertible => 11 and onto
11 => onto => invertible
onto => 11 => invertible