You are Here: Home >< Maths

# Vector Spaces

Announcements Posted on
TSR's new app is coming! Sign up here to try it first >> 17-10-2016
1. A matrix represents a linear transformation with respect to the basis where the matrix is .
If can be represented by a diagonal matrix , find and a basis of with respect to which represents . Also find non-singular matrix such that , where is the matrix representing with respect to .
If and . How would I go about finding the new basis?
2. The new basis is the set of columns of . A matrix is diagonal if and only if it is represented w.r.t. a basis of eigenvectors.
3. (Original post by nuodai)
The new basis is the set of columns of . A matrix is diagonal if and only if it is represented w.r.t. a basis of eigenvectors.
Are the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix would depend on how you write out the eigenvectors?
4. (Original post by JBKProductions)
Are the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix would depend on how you write out the eigenvectors?
The answers are certainly not unique, you can choose any basis of eigenvectors you like. You can try and make it 'more unique' by requiring your basis vectors to be normalized and satisfy some kind of orientation relation -- in fact, in some cases (e.g. Hermitian matrices) you can even take your basis to be orthonormal -- but even then you're still free to change sign and so on.
5. Thanks.
6. (Original post by JBKProductions)
Are the answers(the new basis) not necessarily unique? If the eigenvalues have algebraic multiplicity 2 for example and you can find two linearly independent eigenvectors, then the matrix would depend on how you write out the eigenvectors?
Even if an eigenvalue has multiplicity 1 then there is plenty of choice: Any non-zero vector in a 1-dimensional space is a basis.

## Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank
2. this can't be left blank
3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. Oops, you need to agree to our Ts&Cs to register

Updated: April 16, 2012
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

### How does exam reform affect you?

From GCSE to A level, it's all changing

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read here first

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Study resources

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.