Correspondence Theorem for Vector Spaces - Cooperstein Theorem 2.15

In summary, Theorem 2.15 states that for any surjective function $f: A \to f(A)$, the inverse image of any subset $Y \subseteq f(A)$ is equal to $Y$. However, it is important to note that this is only true for linear transformations, as they ensure the image of a vector space is also a vector space. The proof of part (i) of Theorem 2.15 does not depend on the linearity of the transformation, making it applicable to any function or mapping. Peter raises this point and asks for confirmation, which is given by the other person in the conversation.
  • #1
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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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  • #2
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.
 
  • #3
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
 

FAQ: Correspondence Theorem for Vector Spaces - Cooperstein Theorem 2.15

What is the Correspondence Theorem for Vector Spaces?

The Correspondence Theorem for Vector Spaces, also known as Cooperstein Theorem 2.15, is a fundamental theorem in linear algebra that relates subspaces and quotient spaces of vector spaces. It states that for any vector space V and its subspace U, there is a one-to-one correspondence between subspaces of V containing U and subspaces of V/U, the quotient space of V by U. This theorem is useful in understanding the structure and properties of vector spaces.

How is the Correspondence Theorem for Vector Spaces used?

The Correspondence Theorem for Vector Spaces is used to establish a connection between subspaces and quotient spaces. This connection helps in understanding the structure and properties of vector spaces and is frequently used in proving other theorems in linear algebra. It is also used in applications such as coding theory and differential equations.

What is the significance of Cooperstein Theorem 2.15?

Cooperstein Theorem 2.15, also known as the Correspondence Theorem for Vector Spaces, is significant because it provides a framework for understanding the relationship between subspaces and quotient spaces in vector spaces. It is a fundamental theorem in linear algebra and is used extensively in various areas of mathematics and engineering.

What are the assumptions of the Correspondence Theorem for Vector Spaces?

The Correspondence Theorem for Vector Spaces has two main assumptions: 1) The vector space V is finite-dimensional, and 2) the subspace U is a proper subspace of V. These assumptions are necessary for the theorem to hold and to establish a one-to-one correspondence between subspaces of V containing U and subspaces of V/U.

Can the Correspondence Theorem for Vector Spaces be extended to infinite-dimensional vector spaces?

No, the Correspondence Theorem for Vector Spaces cannot be extended to infinite-dimensional vector spaces. This is because the one-to-one correspondence between subspaces of V containing U and subspaces of V/U only holds for finite-dimensional vector spaces. In infinite-dimensional vector spaces, there are additional complications and the theorem does not hold.

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