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 .
Question
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:
#See solution for details:
Explanation:
-If [tex]w_1[/tex] and [tex]w_2[/tex] are in T(U) then there are [tex]x_1[/tex] and [tex]x_2[/tex] in U so that:
[tex]T(x_)=-w_1\\\\T(x_)=w_2[/tex]
-Then :
[tex]w_1+w_2=T(x_1)+T(x_2)=T(x_1+x_2)[/tex]
-But [tex]x_1+x_2[/tex] is in U since U is a subspace.
-Also, for any scalar c we have:
[tex]cw_1=cT(x_1)=T(cx_1)[/tex], which is in T(U) since [tex]cx_1[/tex] 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.