Can product of two matrices be an isomorphism?

by Mathnewbie   Last Updated November 09, 2018 01:20 AM

Preamble-My knowledge of advanced algebra is very limited, I would appreciate your patience if I misuse the notation. I have tried my best to use the correct notation. In any case, I will be more than happy to edit the question if the mistakes are pointed out to me.

Question- I have two linear transformations $A:\mathcal{V}\rightarrow\mathcal{W}$, and $B:\mathcal{W}\rightarrow\mathcal{V}$, where $\mathcal{V}$ and $\mathcal{W}$ are finite vector spaces over the Filed of real numbers. Then the question is can their product matrix $T=BA$, be an isomorphism?

If $T=BA$ can be an isomorphism what properties should $A$ and $B$ hold?

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