Question

Let V and W be vector spaces, and let T : V \rightarrow WT: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.

Answer 1

By definition,
`T(0)=0 \implies 0 \in T(U)`.

Since
`U` is a subspace, given 2 vectors
`u, v \in U \implies u+v \in U`

`\therefore T(u+v) = T(u)+T(v) \in T(U)`

Then, for some scalar
`\alpha`,

`\alpha T(u) = T(\alpha u) \in T(U)`

since
`\alpha u \in U`. Therefore
`T(U)` is a subspace of
`W` because it satisfies the 3 subspace conditions.