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I have some questions about the concept of subspace of linear transformation and its dimension, when I try to prove following problems:

Prove T is a finite dimensional subspace of L(V) and U is a finite dimensional subspace of V, then

T(U) = {F(u) | F is in T, u is in U} is a subspace of V,

Dim(T(U))<=(dim(T))(dim(U))

What does “T is a finite dimensional subspace of L(V)” mean?

L1(v)+L2(v) = (L1+L2)(v)

aL1(v) = (aL1)(v) ?

and what is dim(T) and dim(T(U))?

Every Linear transformation has its Matrix format, dim(T) is dim(M(T))?

I am a fish in Linear algebra. Hope I explain my questions clearly.

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# Questions about the concept of subspace of linear transformation

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