Let V and W be vector spaces and T:V \rightarrow

dedica66em 2022-01-05 Answered
Let V and W be vector spaces and T:VW be linear. Let {y1,,yk} be a linearly independent subset of R(T). If S={x1,,xk} is chosen so that T(ξ)=yi for i=1,,k, prove that S is linearly independent.
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Thomas Lynn
Answered 2022-01-06 Author has 28 answers
Let us take {w1,w2,w3wk} be the linear independent subset of R(t) instead of {y1yk} and S={v1,v2,vk} is subset of V such that T(vi)=Wi...(1).
The transformation is linear, where V and W are two vector spaces.
To show S={v1,v2,vk} is LI.
For any scalars a1,a2,ak consider a1v1+a2v1++akvk=0...(2)
Since T is a function, the above equation can be written as follows,
T(a1v1+a2v2++akvk)=T(0).
Since T is linear transformation,
T(a1v1)+T(a2v2)++T(akvk)=0
a1T(v1)+a2T(v2)++akT(vk)=0
a1w1+a2w2++akwk=0 (From (1))
Since, the set {w1,w2,w3wk} is LI, the only choice is a1=a2=ak=0...(3)
So, from equation (2) and (3), the set S={v1,v2,vk} is LI.
Hence, proved.
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