Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is from Axler's Linear Algebra Done Right: Chapter 3 Question 14:

Suppose that $W$ is finite dimensional and $T \in L(V,W)$ Prove that if $T$ is injective, then there exists $S \in L(W,V)$ such that $ST$ is the identity map on $V$?

I do not understand why $T$ has to be injective?

For example, why can't just define $S\in L(W,V)$ such that $S(Tv) = v$

and then $(ST)v = S(Tv) = v$?

share|cite|improve this question
Please, try to make the title of your question more informative. E.g., Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. – Julian Kuelshammer Nov 24 '12 at 21:37
ok thanks, ill keep that in mind in the future – mathnoob Nov 24 '12 at 21:38
up vote 2 down vote accepted

If $T$ isn't injective, then there are $u,v\in V$ with $u\ne v$ such that $Tu=Tv$. Then $S(Tu)=S(Tv)$, so you can't have $S(Tu)=u$ and $S(Tv)=v$, since $u\ne v$.

share|cite|improve this answer

If $T$ is not injective, then we can have $Tv_1 = w = Tv_2$, where $v_1 \neq v_2$ and hence $S$ is not well-defined. For such a map, how would you define $Sw$? If you want to define $S(Tv) = v$, then you have the following inconsistency. $$Sw = S(Tv_1) = v_1$$ and also $$Sw = S(Tv_2) = v_2$$

share|cite|improve this answer

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.