The bookUnit 1 · Vector Spaces and Linear Maps22 / 83

Linear transformations, matrix representation, change of basis

Why this is asked: Similar matrices are the same operator in different bases: they share rank, trace, determinant, characteristic and minimal polynomials — but sharing those is not enough to be similar.

Linear transformations, matrices and similarity

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See it move

Change of basis without sign errorsinteractive

Which direction the change-of-basis matrix goes, and the similarity relation that follows.

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The trap here

“AB and BA have the same minimal polynomial” — false

A = [[0,1],[0,0]], B = [[0,0],[0,1]]

AB = A has minimal polynomial $x^{2}$ , BA = 0 has x. The characteristic polynomials do agree.