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$\begingroup$

I pulled this from these notes of lecture 2 from these lectures, see #2 at the top of the notes where it says "$Hom(\mathbb{R}^{n},\mathbb{R}^{n})$ a vector space".

I would like some clarification as to what this means exactly. All elements of $Hom(\mathbb{R}^{n},\mathbb{R}^{n})$ can be represented as $n\times n$ matrices (since it is the set of all linear transformations).

  • What are the scalars of this vector space? $\mathbb{R}$?
  • Can these matrices just be thought of as $\mathbb{R}^{n\times n}$, where these are the elements of the vector space?
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    Yes for both questions. – 2017-01-15
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    @Crostul, thank you very much! :) – 2017-01-15
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    More generally, for any two vector spaces $V,W$ over the same field $F$, the space $\operatorname{Hom}(V,W)$ is also a vector space over $F$. – 2017-01-15

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