12240/Classnotes for Thursday November 8
From Drorbn

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