Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
what have you tried??? –  Praphulla Koushik Oct 21 '13 at 9:05

2 Answers 2

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 $x_1=x_2$.

share|improve this answer

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 $T_2\circ T_1$

share|improve this answer
+1 nice hint :) –  Praphulla Koushik Oct 21 '13 at 9:07

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.