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Suppose $V$ and $W$ are vector spaces with $\dim V = m$ and $\dim W = n$. Show that $L(V,W)$ (the set of all linear transformations from $V$ to $W$) is isomorphic to $\mathbb{R^{n\times m}}$.

How do you typically prove that something is isomorphic to something else? Must it be the case that the set of all linear transformations from $V$ to $W$ is invertible? Alternatively, how would you determine that something is an isomorphism in general?

Hailey
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  • The easiest way to is to determine the dimension of each vector space. In this case both are finite dimensional, so it suffices to show that both spaces have the same dimension. For $L(V,W)$, I believe it has dimension $nm$ and I think you are allowed to say that $\dim \Bbb R^{n \times m} = nm.$ – IAmNoOne Oct 29 '14 at 02:48
  • @Nameless Why does their dimensions being the same mean the two are isomorphic? – Hailey Oct 29 '14 at 03:12
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    The proof deserves to be in another question. Actually you can find online I am sure. The idea is that having the same dimension allows a one-to-one correspondence, I am borrowing most of these concepts from set-theory. – IAmNoOne Oct 29 '14 at 03:16
  • @Hailey see the definition of an isomorphism of vector spaces here. – Ben Grossmann Oct 29 '14 at 03:46

1 Answers1

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Hint 1: To show that two vector spaces are isomorphic, it suffices to show they have the same dimension.

Hint 2: After choosing bases for $V$ and $W$, one may identify $\mathcal{L}(V,W)$ with the vector space of all $n\times m$ real matrices.

Jared
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  • Why does having the same dimension mean that two spaces are isomorphic? – Hailey Oct 29 '14 at 02:59
  • @Hailey: Let's assume that two vector spaces have the same dimension. This means that there is a bijection between any two bases. This bijection can be extended uniquely to a vector space isomorphism. – Jared Oct 29 '14 at 03:24