Correspondence Theorem for Vector Spaces - Cooperstein Theorem 2.15

[Math Amateur]

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ...

I am focused on Section 2.3 The Correspondence and Isomorphism Theorems ... ...

I need help with understanding Theorem 2.15 ...

Theorem 2.15 and its proof read as follows:View attachment 5169It appears to me (and somewhat surprises me) that the proof of part (i) of the above theorem does not seem to depend on the linearity of T and hence would be true for any function/mapping f ...

But is my analysis correct ...

Could someone please confirm that I am correct ... or point out my error(s) ...

Peter

 

[Deveno]

For any surjective function:

$f: A \to f(A)$ it follows that for any subset $Y \subseteq f(A)$ that $f(f^{-1}(Y)) = Y$.

However, it takes a linear transformation to ensure the image of a vector space is again a vector space.

 

[Math Amateur]

For any surjective function:

$f: A \to f(A)$ it follows that for any subset $Y \subseteq f(A)$ that $f(f^{-1}(Y)) = Y$.

However, it takes a linear transformation to ensure the image of a vector space is again a vector space.

Oh! Excellent point ... I had not though of that ...

Thanks for the help ...

Peter