# 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 one-to-one, prove that T is one-to-one. must U also be one-to-one?
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GaceCoect5v
Let UT is one to one.
It is needed to prove that T is one is one.
Take $UT\left(x\right)=0$ this implies $x=0$ since UT is injective.
Similarly,
$T\left(x\right)=0$
$UT\left(x\right)=U\left(0\right)$
$UT\left(x\right)=0$
Hence, T is injective and is one to one.
Therefore, T is one to one.
If $UT\left(x\right)=$0, U may not be one to one as,
$T:{R}^{3}\to {R}^{2}$ defined by $T\left(a.b,c\right)=\left(a.b\right)$.
$T:{R}^{2}\to {R}^{3}$ defined by $T\left(a.b\right)=\left(a,b.0\right)$.
This states that U may not be one to one.