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Suppose that $T_1 : V \to U$ and $T_2 : U \to W$ are linear transformations of vector spaces. If $T_2 \circ T_1$ is one-to-one, prove that $T_1$ is also one-to-one.

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Let for $x_1$ and $x_2$ we have $T_1(x_1)=T_1(x_2)$. So $$T_2\circ T_1(x_1)=T_2(T_1(x_1))=T_2(T_1(x_2))=T_2\circ T_1(x_2)$$ So $x_1=x_2$.

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Hint: suppose $T_1$ is not $1-1$ so there exists $x_1\ne x_2$ with $T_1(x_1)=T_1(x_2)$, now use the injectivity of $T_2\circ T_1$

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+1 nice hint :) – Praphulla Koushik Oct 21 '13 at 9:07

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