Difference between revisions of "12-240/Classnotes for Thursday November 8"

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== Lecture notes upload by [[User:yaaleni.vijay|yaaleni.vijay]] ==
 
 
{{12-240/Navigation}}
 
{{12-240/Navigation}}
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== Riddle Along ==
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Four cars drive in the Sahara desert at constant speeds and with constant directions. A meets B, C, D; B meets C, D. Do C and D meet?
 +
 +
== Goals ==
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1. compute Rank T over A<br>
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2. Compute <math>T^{-1}</math> over <math>A^{-1}</math><br>
 +
3. Solve systems of linear equations
 +
 +
== Theorems ==
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1. Given V' -> V -> W -> W' (where the linear transformations are Q, T, P respectively)<br>
 +
such that P and Q are invertible (i.e. Q is surjective and P is injective)
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then rank T = rank PTQ<br>
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<br>
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2. if T: V -> W, V with basis <math>\beta</math> and W with basis <math>\gamma</math>
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rank <math>[T]_\beta^\gamma</math> = rank T<br>
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<br>
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3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ
 +
 +
== Definitions ==
 +
if A = <math>M_(m \times n)</math>, then it is linear transformation <math>T_A :  F^n -> F^m</math>
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 +
== Lecture notes upload by [[User:yaaleni.vijay|yaaleni.vijay]] ==
 +
  
 
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Latest revision as of 01:15, 4 December 2012

Contents

Riddle Along

Four cars drive in the Sahara desert at constant speeds and with constant directions. A meets B, C, D; B meets C, D. Do C and D meet?

Goals

1. compute Rank T over A
2. Compute T^{-1} over A^{-1}
3. Solve systems of linear equations

Theorems

1. Given V' -> V -> W -> W' (where the linear transformations are Q, T, P respectively)
such that P and Q are invertible (i.e. Q is surjective and P is injective) then rank T = rank PTQ

2. if T: V -> W, V with basis \beta and W with basis \gamma rank [T]_\beta^\gamma = rank T

3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ

Definitions

if A = M_(m \times n), then it is linear transformation T_A :  F^n -> F^m

Lecture notes upload by yaaleni.vijay