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

prsategazd

prsategazd

Answered question

2022-01-05

Let V, W, and Z be vector spaces, and let T:VW and U:WZ be linear.
If UT is onto, prove that U is onto.Must T also be onto?

Answer & Explanation

Marcus Herman

Marcus Herman

Beginner2022-01-06Added 41 answers

Let UT is onto.
It is needed to prove that U is onto.
Let zZ then for some xV.
UT(x)=z but T(x)W.
So U is onto.
But T may not be onto.
Let T:RR2 be T(x)=(x,y) and U:R2R be U(x.y)=0.
Therefore, T may not be onto.

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