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Let $k$ be a field and $V, W$ be two vector spaces which are not necessarily finite-dimensional, I wonder how to find the cardinality of $\text{Hom}_{k}(V,W)$ in terms of the information of $k,V,W$.

I am not sure if this answer is correct.

Any answer or reference will be appreciated!

Arnaud Mortier
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No One
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  • This 23 July 2000 sci.math post (see follow-up here) doesn't address all aspects of your question, but it might be of some use. moments later After posting this comment I saw the comment by @Pedro Tamaroff (now apparently deleted) which led me to realize that I didn't realize you were simply talking about the cardinality of the set (rather than its dimension over $k).$ – Dave L. Renfro Oct 26 '17 at 17:13
  • @DaveL.Renfro No worries, the first paragraph of the only posted answer now makes up for my missing comment. :) – Pedro Oct 26 '17 at 20:11

1 Answers1

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If you have the dimension of $V$ and the cardinality of $W$ available, then $|W|^{\text{dim}(V)}$gives you the cardinality of $\text{Hom}_k(V,W)$, because a linear map from $V$ to $W$ is given by arbitrarily mapping a basis for $V$ into $W$ and then extending linearly.

If you have only the dimensions of $V$ and $W$, then, in order to apply the preceding information, you need to find the cardinality of $W$. This will be $|k|^{\text{dim}(W)}$ if $W$ is finite-dimensional. If, on the other hand $\text{dim}(W)$ is infinite, then $|W|$ will be $\text{dim}(W)\cdot |k|^{<\omega}$, where $|k|^{<\omega}$ means the number of finite sequences of elements of $k$, which equals $|k|$ if $k$ is infinite and $\aleph_0$ if $k$ is finite.

Andreas Blass
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