Difference between revisions of "12240/Classnotes for Thursday November 8"
From Drorbn
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{{12240/Navigation}}  {{12240/Navigation}}  
+  == 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<br>  
+  2. Compute <math>T^{1}</math> over <math>A^{1}</math><br>  
+  3. Solve systems of linear equations  
+  
+  == Theorems ==  
+  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)  
+  then rank T = rank PTQ<br>  
+  <br>  
+  2. if T: V > W, V with basis <math>\beta</math> and W with basis <math>\gamma</math>  
+  rank <math>[T]_\beta^\gamma</math> = rank T<br>  
+  <br>  
+  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>  
+  
+  == Lecture notes upload by [[User:yaaleni.vijayyaaleni.vijay]] ==  
+  
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−  Image:12240 Nov 8 Page  +  Image:12240 Nov 8 Page 1.jpgPage 1 
Image:12240 Nov 8 Page 2.jpgPage 2  Image:12240 Nov 8 Page 2.jpgPage 2  
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Latest revision as of 02:15, 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 over
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 and W with basis
rank = rank T
3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ
Definitions
if A = , then it is linear transformation