Let V and W be vector spaces, and let T W V ! W be a linear transformation. Given a subspace U of V , let T .U / denote the set of all images of the form T .x/, where x is in U. Show that T .U / is a subspace of W .

#See solution for details:

Explanation:

-If and are in T(U)  then there are and in U so that:

-Then :

-But is in U since U is a subspace.

-Also, for any scalar c we have:

, which is in T(U) since is in U

#Finally, 0=T(0) so 0 is in T(U) thus T(U) is a subspace of W as it satisfies the above 3 subspace conditions.

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