vector spaces and linear maps

Suppose that V and W are vector spaces. Let Hom (V, W ) be the set of all

linear maps φ : V → W . Add and scale linear maps in the obvious way to

make Hom (V, W ) into a vector space. Deﬁne a linear map

and prove that your map is an isomorphism.

Hom (Rp , Rq ) → Rpq ,

HELP PLEASE!!!!!!!!!

Re: vector spaces and linear maps

You know, I presume, that, given bases for V and W, any linear map from V to W can be written as a matrix. So this question is asking you to construct a linear map from the set of p by q matrices (which each have pq numbers, of course) to the set of ordered "pq-tuples". There is an obvious way to do that: for example, mapping to . Try it.

Re: vector spaces and linear maps

specifically, since we know that are vector spaces, for any two bases we can form, for any , the matrix . explicitly, if , and , then the columns of are:

are usually (but not always) taken to be the standard bases, in which case we can drop the brackets and simply write: for the j-th column of .

so the desired isomorphism is:

note that there is no special property of used above, and we have exactly the same argument for any two vector spaces U,V with dim(U) = p, and dim(V) = q, over an arbitrary field F, that is:

where the second isomorphism is just the "vectorization of a matrix", given by concatenating the columns.