Could you please help me with this problem?
Prove that a matrix associated with an endomorfism has inverse if and only if is injective.
Thanks a lotĦĦ
For instance, if you go for the obvious property, is invertible if and only if there exists a matrix such that . Then, suppose is not injective. that means there exists some non-zero vector which is mapped to zero (that is to say, ). Call this vector . Then but 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.