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Linear transformation with a vector problem

  • 29-07-2010 3:57pm
    #1
    Registered Users, Registered Users 2 Posts: 1,666 ✭✭✭


    Hi I was hoping someone could point me in the right direction with a question I have been looking at, I don't have any notes for this section of my course.

    I have been looking around on the internet for examples but am having little luck probably as I am not searching for the correct terms.

    The question is Q1 part (b)

    2010page1algebra.jpg

    I'm not sure how to proceed at all really, if someone could even just give me a link to something appropriate that would be brilliant. I have a reasonable of understanding of elementary matrices if that is any help.

    Thanks for taking the time :)


Comments

  • Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    The question basically comes down to solving a pair of linear simultaneous equations. You're told that some vector (x,y) maps to (x+3y, 2x+4y), now you have to find what vector (x,y) maps to (0,1), take them seperately:

    x -> x + 3y = 0
    y -> 2x + 4y = 1

    two equations, two variables, try solving it and then check your answer, which is easy.


  • Registered Users, Registered Users 2 Posts: 1,666 ✭✭✭charlie_says


    Thanks for the response.

    Stuck on something else now but making progress overall :)


  • Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    What answer did ya get? :D


  • Registered Users, Registered Users 2 Posts: 1,666 ✭✭✭charlie_says


    v=(3/2,-1/2) was what I got :)

    Moving on the the next question 2 (a), I was wondering if I could pick your brains again:

    I calculated AB correctly and I got out B^-1 right, but the next part to solve the system, is this just a straight guassian elimination to solve or does it relate to the identity matrix I got for AB or my answer for B^-1?

    Thanks again :)


  • Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    You can obviously do it by Gaussian Elimination but yeah, they want you to use the result you got for B^-1. You should be able to rewrite the system of linear equations in matrix from. When you have this done note that the 3x3 matrix comes out as exactly B. Since you have B^-1 already, solving it becomes very simple (just comes down to a multiplication of a 3x3 matrix by a vector)...


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  • Registered Users, Registered Users 2 Posts: 1,666 ✭✭✭charlie_says


    I multiplied B^-1 by the vector (-2,1,-5) to get:

    x= -7/3
    y= -2
    and z=-5/3

    Thanks again for your help Michael

    For the b part where I have to find the inverse of C, I am going to use the theorem

    C^-1 = (1/det(C)).(adj(C))


  • Moderators, Science, Health & Environment Moderators Posts: 1,852 Mod ✭✭✭✭Michael Collins


    You're welcome. Yeah, pretty much any inverse method will work fine.

    (All your answers have been right so far by the way.)


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