Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta

Diagonalizable matrix used in polynomial form? watch

    • Thread Starter


    If anyone you help with the last part of c i'd be very grateful!

    What does the first part of (c) tell you about the span

    <B^n, B^(n-1), ... , B^2, B, I>?

    (PS the Oxford mods papers are online and may be easier to link to directly rather than producing a scan)

    The idea is to use the identity from the first part of c) to re-write \displaystyle \Sigma a_i B^i in a more helpful form.
    • Thread Starter

    Hmm.. I must be being rather slow, still not really getting it. The first part of c tells us that the span <B^n, B^(n-1), ... , B^2, B, I> can equal zero for certain values of a[i] not equal to zero, so B^i are dependent? Don't know if that's helpful. Not sure about rewriting the span using the identity from the first bit, sorry!

    Oh wait, if B^i are linearly Dependant then they span the 3x3 matrices so there exists a linear combination that creates the required matrix?

    If B^2 = (\lambda_1+\lambda_2)B - \lambda_1 \lambda_2 I, then what can be said about the set \{I,B,B^2\}, with regards to linear independence/dependence?

    Also, note that we cannot have \{I,B,B^2,\ldots,B^n\} being a linearly independent set for n \geq 9 (irrespective of what B actually is), since the dimension of the space of 3 x 3 matrices is 9, and a linearly independent set cannot have more elements than the dimension of the space it is in.

    So getting back to the question - it's possible to write the LHS of the equation at the end of C entirely in terms of B and I, can you see why?
Submit reply
Turn on thread page Beta
Updated: March 31, 2011
Do you agree with the proposed ban on plastic straws and cotton buds?
Useful resources

Make your revision easier


Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here


How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

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

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