# Can product of two matrices be an isomorphism?

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

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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