I am sorry for this basic question:
let We have two vector spaces such that one of them is finite dimensional and another one is infinte dimensional. I want to know whether I can define a linear map between them? I want to take a finite basis for infinite dimensional vector space and then define the map.
[Math] map between finite and infinite vector spaces
linear algebra
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Best Answer
Yes you surely can. But linear maps correspond to matrices only if the corresponding vector spaces are finite dimensional.
And no, you cannot take a finite basis for an infinite dimensional vector space, but you can choose finitely many linearly independent vectors and extend them to a basis (using zorns lemma or something, to prove this extension existence) and define the map on those finitely many vectors by some image and define the map to be zero on the other vectors. you also can define the image for an infinite basis in general (in particular this is done above).