prsategazd

2022-01-05

Let V, W, and Z be vector spaces, and let $T:V\to W$ and $U:W\to Z$ be linear.

If UT is onto, prove that U is onto.Must T also be onto?

If UT is onto, prove that U is onto.Must T also be onto?

Marcus Herman

Beginner2022-01-06Added 41 answers

Let UT is onto.

It is needed to prove that U is onto.

Let$z\in Z$ then for some $x\in V$ .

$UT\left(x\right)=z$ but $T\left(x\right)\in W$ .

So U is onto.

But T may not be onto.

Let$T:R\to {R}^{2}$ be $T\left(x\right)=(x,y)$ and $U:{R}^{2}\to R$ be $U(x.y)=0$ .

Therefore, T may not be onto.

It is needed to prove that U is onto.

Let

So U is onto.

But T may not be onto.

Let

Therefore, T may not be onto.

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