Let V and W be vector spaces, let T: V --> W be linear,and let {w 1 , w 2 , ...,

Question

Let V and W be vector spaces, let T: V --> W be linear,and let {w1, w2, ..., wk} be alinearly independent subset of R(T). Prove that if S ={v1,v2, ...,vk} is chosen so thatT(vi) = wi for i = 1, 2, ..., k, then S islinearly independent.

Explanation / Answer

Suppose S = {v1,v2, ...,vk}is not linearly independent. Then there exists scalars c1,c2,...,ck not all zero such that:    c1v1 +c2v2 + ... + ckvk =0     =>   T(c1v1 + c2v2 + ... + ckvk ) = T(0) = 0 but since T is linear , we get :     c1T(v1) +c2T(v2) ... +ckT(vk ) = 0 or, c1w1 +c2w2 ... + ckwk= 0 which implies that {w1,w2,...,wk} is not linearly independent since the scalarsc1,c2, ...,ck are not allzero. But, then we get a contradiction since{w1,w2, ...,wk} is a linearlyindependent set. Hence, there exists no scalarsc1,c2, ...,ck not allzero such that :    c1v1 +c2v2 + ... + ckvk =0   Therefore, S = {v1,v2,...,vk} is linearly independent.

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