b2sonicxh

2022-01-07

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

If U and T are one-to-one and onto, prove that UT is also

If U and T are one-to-one and onto, prove that UT is also

Navreaiw

Beginner2022-01-08Added 34 answers

Let U and T is one to one.

Assume,$UT\left(x\right)=UT\left(y\right)$

$T\left(x\right)=T\left(y\right)$ U is one to one

$x=y$ T is one to one

So, if U and T is one to one, then UT is also one to one.

Suppose U and T is onto, then by definition of onto$T\left(x\right)=y$ , for all y W and

$U\left(y\right)=z$ for all $z\in Z$ .

$UT\left(x\right)=UT\left(y\right)$

$=2$

So, that UT is onto.

Hence, UT is onto.

Assume,

So, if U and T is one to one, then UT is also one to one.

Suppose U and T is onto, then by definition of onto

So, that UT is onto.

Hence, UT is onto.

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