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

Joyce Smith 2022-01-04 Answered
Let V and W be vector spaces, let T:VW 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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Expert Answer

Daniel Cormack
Answered 2022-01-05 Author has 34 answers

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:VW 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,βF and u,vV
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 αiFS for i=1,2,,k
Since T is linear, 0vV will always gets mapped to 0wW
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.

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