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# Linear Algebra Dimension Question watch

1. If hom(V,W) denotes the space of linear transformations from V to W, why does dim(hom(V,W))=dim(V)*dim(W)?

Also, if dim(V)=dim(W) and T:V->W is 1-1, why must is also be onto? Does it work to say that the image of a basis of V is a basis of W (from the kernel being {0}) and then using linearity to cover all of W?

Thanks
2. (Original post by james22)
If hom(V,W) denotes the space of linear transformations from V to W, why does dim(hom(V,W))=dim(V)*dim(W)?

Also, if dim(V)=dim(W) and T:V->W is 1-1, why must is also be onto? Does it work to say that the image of a basis of V is a basis of W (from the kernel being {0}) and then using linearity to cover all of W?
It does follow from ker T = {0} that the image of a basis in V is a basis of W, but it is not entirely trivial and should be proved.

That this implies dim W = dim V is immediate (and doesn't really need even a mention of linearity).

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Updated: November 18, 2013
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