Theoretical linear algebra final in 14 hours

In summary: I have some fundamental questions to ask. however, I can only think of two right now.In summary, a linear transformation is invertible if and only if its representing matrix is invertible. If a linear transformation is invertible, then it is also an isomorphism. The relation between the eigenspace and the kernel of a square matrix is that if the matrix is invertible, the kernel is just the 0 vector, but if the matrix is not invertible, the kernel is the eigenspace of the 0 eigenvalue.
  • #1
yaganon
17
0
**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...
 
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  • #2


yaganon said:
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.
 
  • #3


yaganon said:
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...
 

FAQ: Theoretical linear algebra final in 14 hours

1. What is theoretical linear algebra?

Theoretical linear algebra is a branch of mathematics that deals with the study of vector spaces and linear transformations. It involves the use of matrices and other mathematical tools to understand and solve problems related to systems of linear equations, vector spaces, and linear transformations.

2. What topics are typically covered in a theoretical linear algebra final?

A theoretical linear algebra final may cover topics such as vector spaces, linear independence and bases, linear transformations, eigenvalues and eigenvectors, inner product spaces, and diagonalization. It may also include applications of linear algebra in other fields such as physics and engineering.

3. How can I prepare for a theoretical linear algebra final in 14 hours?

Preparing for a theoretical linear algebra final in 14 hours can be challenging, but there are some strategies you can use. First, review your notes and class materials to identify key concepts and equations. Next, work through practice problems to test your understanding and identify any areas that need further review. Finally, make sure to get enough rest and stay hydrated to keep your mind sharp during the exam.

4. What are some common mistakes to avoid in a theoretical linear algebra final?

Some common mistakes to avoid in a theoretical linear algebra final include not fully understanding the definitions and properties of vector spaces, not being familiar with the properties of matrices, and not properly applying the rules of matrix operations. It is also important to carefully read and understand the questions and to double-check your work for any errors.

5. What are the real-world applications of theoretical linear algebra?

Theoretical linear algebra has many real-world applications, including computer graphics, data analysis, cryptography, and physics. It is used in creating 3D animations, analyzing large datasets, encoding and decoding information, and modeling physical systems. It is also a fundamental tool in many branches of engineering and science.

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