Question

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 .

Answer 1

Answer:

#See solution for details:

Explanation:

-If $w_1$ and $w_2$ are in T(U)  then there are $x_1$ and $x_2$ in U so that:

$T(x_)=-w_1\\\\T(x_)=w_2$

-Then :

$w_1+w_2=T(x_1)+T(x_2)=T(x_1+x_2)$

-But $x_1+x_2$ is in U since U is a subspace.

-Also, for any scalar c we have:

$cw_1=cT(x_1)=T(cx_1)$, which is in T(U) since $cx_1$ 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.