12-240/Classnotes for Thursday November 8

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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