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

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?
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Marcus Herman
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 $T\left(x\right)=\left(x,y\right)$ and $U:{R}^{2}\to R$ be $U\left(x.y\right)=0$.
Therefore, T may not be onto.