Prove the relation T = S.

CoormaBak9 2020-12-31 Answered
Prove the relation T=S.
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Expert Answer

Bertha Stark
Answered 2021-01-01 Author has 96 answers
The vector space B={v1,,vk} is a basis for vector space V, and S:VW,T:VW are linear transformations such that S(vi)=T(vi).
Suppose vector v in V.
v=a1v1+a2v2++anvn
where, a1, a2,  an are scalars.
Suppose, S(v)=S(a1v1+a2v2++anvn).(1)
As S is a linear transformation therefore,
S(v1+v2)=S(v1)+S(v2)
S(av)=aS(v)
S(v)=S(a1v1+a2v2++anvn)
=S(a1v1)+S(a2v2)++S(anvn)
=a1S(v1)+a2S(v2)++anS(vn)
Also, S(vi)=T(vi)
So,
S(v)=a1S(v1)+a2S(v2)++anS(vn)
=a1T(v1)+a2T(v2)++anT(vn)
=T(a1v1+a2v2++anvn)
=T(v)
As, v in V is arbitrary.
S(v)=T(v) for all v in V.
This proves that S=T.
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