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How do you get a matrix from this basis?

  1. Apr 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Here's my problem. I only need help with the bottom part, but if you could explain the problem more vividly that would help too.
    Kwvdeca.png
    2. Relevant equations
    A = S-1BS (?)
    There aren't really any relevant equations. This part of linear algebra is getting really abstract, at least I think so. If there are any equations, I'm not aware of them.
    3. The attempt at a solution
    The problem I'm assigned for homework is very similar to this. I'm almost sure I got the answer right, but I don't completely understand the question. Why does the basis give that specific matrix B? What do you do with the basis to get B?
     
  2. jcsd
  3. Apr 30, 2015 #2

    BvU

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    Hello BS, welcome to PF :smile: !

    Your hitting an extremely important subject in physics, math, engineering and what have you here !

    Unfortunately, the explanation given in your text is already quite clear, so I find it difficult to add to it meaningfully.

    Do you have any problem with the interpretation of matrix B on the basis of ##\mathfrak {B}## ?

    How do you write vectors that are written as ##(\lambda, \mu)## on the basis of ##\mathfrak {B}## if you have to write them on the usual basis on ##\Re^2## ?

    And if you apply the transformation B to ##(\lambda, \mu)## what do you get? And how does it look in ##\Re^2## ?

    The other way around: what do you do to find the ##(\lambda, \mu)## corresponding to a vector ##(x, y)## in ##\Re^2## ?

    ---
    Sorry about the ##\Re##, looks awful. Someone will know which fonts are available here, but I have trouble finding out

    [edit] ah, found \mathbb : ##\mathbb R##
    --​
     
    Last edited: Apr 30, 2015
  4. Apr 30, 2015 #3

    BvU

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    Perhaps I can save a few words with a picture:

    VecTransf.jpg
    T is reflection about the line spanned by (2,3).
    What is T of vector ##\vec v = (5,2)## ?

    We found that T is diagonal on a basis ##\mathfrak B## consisting of ##\vec e_1 = (2,3)## and ##\vec e_2 = (-3,2)##
    with ##T(\vec e_1) = (1) \vec e_1## and ##T(\vec e_2) = (-1) \vec e_1##
    T is linear, which means that
    the projection ##\lambda \vec e_1## of ##\vec v## on ##\vec e_1##
    transforms into ##\lambda T(\vec e_1) = \lambda \vec e_1##

    and the projection ##\mu \vec e_2## of ##\vec v## on ##\vec e_2##
    transforms into ##\mu T(\vec e_2) = - \mu \vec e_2##
     
  5. May 1, 2015 #4

    Fredrik

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    The formula for the ##\mathfrak B##-matrix of ##T## (for arbitrary ##\mathfrak B##) is certainly relevant.
     
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