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

Let V, W, and Z be vector spaces, and let $$\displaystyle{T}:{V}\rightarrow{W}$$ and $$\displaystyle{U}:{W}\rightarrow{Z}$$ be linear.
If U and T are one-to-one and onto, prove that UT is also

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Navreaiw
Let U and T is one to one.
Assume, $$\displaystyle{U}{T}{\left({x}\right)}={U}{T}{\left({y}\right)}$$
$$\displaystyle{T}{\left({x}\right)}={T}{\left({y}\right)}$$ U is one to one
$$\displaystyle{x}={y}$$ 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 $$\displaystyle{T}{\left({x}\right)}={y}$$, for all y W and
$$\displaystyle{U}{\left({y}\right)}={z}$$ for all $$\displaystyle{z}\in{Z}$$.
$$\displaystyle{U}{T}{\left({x}\right)}={U}{T}{\left({y}\right)}$$
$$\displaystyle={2}$$
So, that UT is onto.
Hence, UT is onto.