Theoretical linear algebra final in 14 hours

[yaganon]

**theoretical linear algebra final in 14 hours***

I have some fundamental questions to ask.1. a linear transformation is invertible iff A is invertible. IS THIS TRUE? (dumb question but my brain hurts and I just need confirmation)

2. if a linear transformation is invertible, then it's an isomorphism RIGHT?

3. what is the relation between the eigenspace and the kernel of any given square matrix?

P.S. I like it better if you answer like this:
1. YES
2. YES
3. well...

 

[JasonRox]

I have some fundamental questions to ask. however, I can only think of two right now.


a linear transformation is invertible iff A is invertible. IS THIS TRUE? (dumb question but my brain hurts and I just need confirmation)

what is the relation between the eigenspace and the kernel of any given square matrix?

Linear transformation has inverse implies that it is one-to-one, and if represented as a matrix, it does not necessarily need to be invertible.

Take for example the following linear transformation from R^2 to R^3... that is (x,y) goes to (x,y,0). This matrix will not even be square. So, it can NOT be invertible of course. But it is one-to-one.

But if A is an invertible matrix, then it does represent a linear transformation that has an inverse.

 

[HallsofIvy]

I have some fundamental questions to ask.


1. a linear transformation is invertible iff A is invertible. IS THIS TRUE? (dumb question but my brain hurts and I just need confirmation)

What is "A"?? The matrix representing the linear transformation? If so, yes, of course, each is invertible if and only if the other is. That pretty much goes with saying that the matrix "represents" the linear transformation.

2. if a linear transformation is invertible, then it's an isomorphism RIGHT?

Yes.

3. what is the relation between the eigenspace and the kernel of any given square matrix?

I know what the eigenspace of a given eigenvalue of a matrix is but I don't know what you mean by "eigenspace of a matrix". The space of all eigenvectors of the matrix?

If a matrix is invertible, then its kernel is just the 0 vector. Otherwise 0 is an eigenvalue and the kernel is the eigenspace of the 0 eigenvalue. If you mean "space of all eigenvectors of the matrix" then I think the best you can say is that "the kernel is always a subspace of the eigenspace".

P.S. I like it better if you answer like this:
1. YES
2. YES
3. well...