The last bullet point on page 60 is wrong

[razeghi71]

Describe the mistake The last consequence that has been listed at the end of page 60 is that: if $dim(v) = dim(w)$ then $\Phi$ is injective and surjective and bijective. but I think the right version should be: If $dim(v) = dim(Im(\Phi))$ then $\Phi$ is injective and surjective and bijective.

Location Please provide the

1. 2023-10-18
2. 2.7.3
3. 60
4. descriptions under formula (2.129)

Proposed solution Change $dim(v) = dim(w)$ to $dim(v) = dim(Im(\Phi))$

[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