Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    Given that M =

    4 -5
    6 -9

    and that the eigenvalues of M are 1 and -6

    A transformation T:R^2 -> R^2 is represented by the matrix M. There is a line through the origin for which every point on the line is mapped onto itself under T.

    How do I find a cartesian equation of the line

    The mark scheme says “stating, implying or showing λ = 1 associated with point invarient line). Why is this? Why shouldn’t λ = -6...............

    ??
    Offline

    10
    ReputationRep:
    You want Mv = v, which means the eigenvalue (the coefficient of v on the RHS) should be 1.
    Offline

    15
    ReputationRep:
    It might help to think in the following way:
    The line has equation y=kx (as it passes through the origin. Hence every point on the line has the general coordinate (x,kx).
    If every point is mapped to itself then we have:
    \left(

\begin{array} {c c}

4 & -5 \\

6 & -9

\end{array}

\right)

\left(

\begin{array}{c}

x \\

kx

\end{array}

\right)

=

\left(

\begin{array}{c}

x \\

kx

\end{array}

\right)
    That is:
    

\left(

\begin{array}{c}

4x-5kx \\

6x-9kx

\end{array}

\right)

=

\left(

\begin{array}{c}

x \\

kx

\end{array}

\right)
    Dividing both sides by x:
    

\left(

\begin{array}{c}

4-5k \\

6-9k

\end{array}

\right)

=

\left(

\begin{array}{c}

1 \\

k

\end{array}

\right)

    Clearly k=\frac{3}{5} so the line has equation y=\frac{3}{5}x
 
 
 
Turn on thread page Beta
Updated: June 27, 2005
The home of Results and Clearing

2,416

people online now

1,567,000

students helped last year

University open days

  1. Keele University
    General Open Day Undergraduate
    Sun, 19 Aug '18
  2. University of Melbourne
    Open Day Undergraduate
    Sun, 19 Aug '18
  3. Sheffield Hallam University
    City Campus Undergraduate
    Tue, 21 Aug '18
Poll
A-level students - how do you feel about your results?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

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