Hello
Could you please help me with this problem?
Prove that a matrix associated with an endomorfism $\displaystyle T$ has inverse if and only if $\displaystyle T$ is injective.
Thanks a lotĦĦ
Well, a matrix $\displaystyle T$ has an inverse if and only if...what? There are so many properties that finish off this sentence, but pick your favourite. Does this property tell you anything about the matrix begin injective? Surjective?
For instance, if you go for the obvious property, $\displaystyle T$ is invertible if and only if there exists a matrix $\displaystyle S$ such that $\displaystyle ST=I=TS$. Then, suppose $\displaystyle T$ is not injective. that means there exists some non-zero vector which is mapped to zero (that is to say, $\displaystyle ker(T) \neq \{0\}$). Call this vector $\displaystyle v$. Then $\displaystyle vI=v$ but $\displaystyle vTS = 0S=0$ an contradiction. Thus the matrix must be injective.
Now, suppose that your matrix is an injective endomorphism and prove that this implies that it is invertible.