MHB Correspondence Theorem for Vector Spaces - Cooperstein Theorem 2.15

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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
 
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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.
 
Deveno said:
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
 
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This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
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