12-240/Classnotes for Thursday November 8
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 [math]\displaystyle{ T^(-1) }[/math] over [math]\displaystyle{ A^(-1) }[/math] 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 [math]\displaystyle{ \beta }[/math] and W with basis [math]\displaystyle{ \gamma }[/math]
rank [math]\displaystyle{ [T]_\beta^\gamma }[/math] = rank T
3. if P and Q are invertible matrices, A is some other matrix, rank A = rank PAQ
Definitions
if A = [math]\displaystyle{ M_(m \times n) }[/math], then it is linear transformation [math]\displaystyle{ T_A : F^n -\gt F^m }[/math]
Lecture notes upload by yaaleni.vijay
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