WTS: Hom(V,W) which is the set of all linear maps is a vector space.
The question I have is that, isn't Hom(V,W) a subset of W? Because Hom(V,W) is just the set of all linear maps from V to W, so why can't I just show that it is a subspace?
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2A map from $V\to W$ is not an element of $W$. – pancini Mar 14 '19 at 23:43
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Because a linear map is not a vector. – Bernard Mar 14 '19 at 23:43
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3@Bernard A linear map is very capable of being a vector, it being a member of the vector space $\mathrm{Hom}(V,W)$! – Joseph Martin Mar 14 '19 at 23:54
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1I mean not a vector of the vector spaces at hand ($V$ and $W$). – Bernard Mar 14 '19 at 23:57
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Hom(V,W) is a set of homomorphisms(vector space homomorphism)/linear maps from V to W, which forms vector space! whilst what you're thinking is just image of V with particular element f of Hom(V,W)(i.e. f(V) is subspace of W). These two are different things!
Devyani
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You are probably thinking of the image of a particular function $V\to W$. By itself, a function $f\colon V\to W$ is a single object. You can define a vector space structure on the set of all such maps since it contains a zero element (the zero map) and you can scale any linear map by a constant: if $f\colon V\to W$ is linear and $c$ is a scalar, then $cf$ defined by $(cf)(v)=c(f(v))$ is another linear map.
pancini
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1That's not enough for a vector space, you also need addition of linear maps (which, of course, works the same way), – celtschk Nov 11 '19 at 08:32