The last bullet point on page 60 is wrong

[razeghi71]

The last bullet point on page 60 is wrong

[mpd37]

I believe the statement is correct as it is: If dim(V) = dim(W) then it holds that

injective <=> surjective <=> bijective

Assume dim(V) = 1, dim(Im(\Phi)) = 1, dim (W) = 2

We can have an injective mapping, but that wouldn't be surjective or bijective.

I may miss something, though.

[razeghi71]

That's true, what I wrote instead was also wrong, but lets say $V = W = R^2$ and $\Phi$ is a function that maps all vectors in $V$ say $[x, y]^T$ to $[x, 0]^T$ in $W$, here $Im(\Phi) \subseteq W$ but $\Phi$ is not surjective, injective.

[mpd37]

OK. Then I'll close this issue