Let V, W, and Z be vector spaces, and let

Nicontio1 2022-01-04 Answered
Let V, W, and Z be vector spaces, and let T:VW and U:WZ be linear.
If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?
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Expert Answer

GaceCoect5v
Answered 2022-01-05 Author has 26 answers
Let UT is one to one.
It is needed to prove that T is one is one.
Take UT(x)=0 this implies x=0 since UT is injective.
Similarly,
T(x)=0
UT(x)=U(0)
UT(x)=0
Hence, T is injective and is one to one.
Therefore, T is one to one.
If UT(x)=0, U may not be one to one as,
T:R3R2 defined by T(a.b,c)=(a.b).
T:R2R3 defined by T(a.b)=(a,b.0).
This states that U may not be one to one.
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