I presume you mean . You don't "prove it"- that is the definion of T(X)
let be a linear mapping from vector spaces U to V and let X be a subspace of U. Show that
is a subspace of V
I can understand why this is, it seems pretty trivial but I am not sure how you would go about proving it.
Thanks for any help
I think you meant
right? (Note the capital X on the LHS.)
I think I'd need a bit more background in order to understand your problem, because this is probably how I would define the set on the LHS. How does your book or professor define the LHS normally?
Suppose u and v are in T(X). That is, there exist x in X such that u= T(x) and there exist y in X such that v= T(y). Now u+ v= T(x)+ T(y)= T(x+ y).
Suppose u is in T(X) and a is any scalar. Then there exist x in X such that u=T(x). Now au= aT(x)= T(ax).
Do you see how those prove that u+ v and au are in T(X)?