Let V and W be vector spaces, let T:V→W be linear, and let {w1,w2,…,wk} be a linearly independent set of k vectors from R(T). Prove that if S={v1,v2,...,vk} is chosen so that T(vi)=Wi for i=1,2,…,k, then S is linearly independent.

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Daniel Cormack · Accepted Answer

Definitions: 1) The set of vectors {v1,v2,...,vk} from a vector space V is said to be linearly independent if the linear combination of vectors α1v1+α2v2+…+αkvk=0 ⇒α1=α2=…=αk=0 2) Let V and W be vector spaces over the same field F. The map T:V→W is said to be linear transformation from V to W if (i) T(0v)=0w (ii) T(αu+βv)=αT(u)+βT(v) where α,β∈F and u,v∈V Let us consider an arbitrary representation of zero vector of V as a linear combination of vectors from S. 0v=α1v1+α2v2+…+αkvk=∑i=1kαivi...(1) where αi∈FS for i=1,2,…,k Since T is linear, 0v∈V will always gets mapped to 0w∈W i.e., T(0v)=0w Thus, 0w=T(0v)=T(∑i=1kαivi) By the linearity of T we have 0w=T(∑i=1kαivi)=∑i=1kαiT(vi) ∑i=1kαiwi (Since T(vi)=wi) =α1w1+α2w2+…+αkwk since the set {w1,w2,…,wk} is linearly independent ⇒α1=α2=…=αk=0 substituting α1=α2=...=αk=0 in (1) we have, the set {v1,v2,...,vk} are linearly independent. Thus (1) will be the trivial representation of zero vector of V. Hence S={v1,v2,...,vk} is linearly independent.

Let V and W be vector spaces, let T : V \rightarrow W be linear, and let \{w_1 , w_2 , ...

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