In Problem 4 we use the following notation. For any m×n matrix A, let TA be the

Question

In Problem 4 we use the following notation. For any m×n matrix A, let TA be the linear transformation from R n to R m defined by TA(~x) = A~x for all ~x ? R n

Problem 4. Let A nd be nxn matrices Problem 4. Let A and B be n × n matrices. (a) Assume that A CB for some matrix C. Prove that if C is invertible, then ker(TA)ker(TB). (b) Prove that if AB - BA, then ker(TA) and ker(TB) are both contained in ker(TAB). Give an example of matrices A and B such that ker(TA) and ker(TB) are proper subsets of ker(TAB).

Explanation / Answer

Ker{TAC}=ker{TBC}

I mean

There is given A=CB so that is inverts above answer is correct

2. Simple

{TAB}={TBA} those are unity by the Ta and Tb

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