let $V(F^n,F^m)$ be the vector space of all linear transformations from $F^n$ to $F^m$. prove that it is isomorphic to $M_{m\times n}$.
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2026-03-26 17:34:38.1774546478
isomorphism of vector spaces of linear transformation and $m\times n$ matrices
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First you need to think of an isomorphism. Say $\{e_i\}$ is a basis for $F^n$. Define $g:V(F^N,F^M) \longrightarrow M_{m\times n}$ by $g(v) =[v(e_1),...,v(e_n)]$ for $v\in V(F^N,F^M)$ You should be able to show that g is linear and bijective so is an isomorphism.