The following true/false question was posed:

An isomorphism between to vector spaces can always be represented by a square singular matrix.

This is not true. I know that in the case of finite dimensional vector spaces:

A map is an isomorphism iff it can be represented by a square non-singular matrix.

I would like to know whether it is possible to prove this in general (i.e for all vector spaces)?


In general, an isomorphism between vector spaces can't even be represented by a matrix. Take, for example the vector space $V$ of all functions $\mathbb{R} \to \mathbb{R}$ This vector space is isomorph with itself by $I:V\to V: f(x) \to f(x+1) $ But there is no finite or even countable base, so there is no matrix representaion of $I$


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