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LINEAR ALGEBRA: image of vectors through other basis

  1. Jun 2, 2013 #1
    1. The problem statement, all variables and given/known data

    In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)##
    and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where
    (1 0 0)
    (0 2 0) = A
    (0 0 0)
    find:
    ##\phi(e_1), \phi(e_2), \phi(e_3)## written with respect to the canonical basis
    (where e1, e2, e3 are vector of the canonical basis ##\mathcal{E}##)


    3. The attempt at a solution

    here is what I thought, but having no solutions i don't know if it is correct:

    I write the vectors of the canonical basis as combination of the vectors of B, also obtaining the ##M^{B,\mathcal{E}}##:

    (1,0,0)=##a_{11} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{21} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{31} \cdot (0,0,1)##

    (0,1,0)=##a_{12} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{22} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{32} \cdot (0,0,1)##

    (0,0,1)=##a_{13} \cdot \frac{1}{\sqrt{2}}(1,1,0)+ a_{23} \cdot \frac{1}{\sqrt{2}}(1,-1,0)+ a_{33} \cdot (0,0,1)##

    getting:

    (##\frac{\sqrt{2}}{2}## ##\frac{\sqrt{2}}{2}## 0)
    (##\frac{\sqrt{2}}{2}## -##\frac{\sqrt{2}}{2}## 0)= ##M^{B, \mathcal{E}}##
    ( 0 0 1)

    so ##\phi(e_1), \phi(e_2), \phi(e_3)## are the columns of this matrix.
    Is it correct?
    thank you in advance :)
     
  2. jcsd
  3. Jun 2, 2013 #2

    Fredrik

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    I haven't thought it all through, but it doesn't look correct to me. I don't see why those columns would contain the ##\mathcal E##-components of the ##\phi(e_i)##. Do you have an argument for it? Also, I think I have calculated ##\phi(e_3)##, and my result is different from yours.

    I think the strategy here should be to start with
    $$\phi(e_i)=\phi\left( \sum_j M^{B,\mathcal E}_{ji} v_j\right) =\sum_j M^{B,\mathcal E}_{ji}\phi(v_i).$$ Now you just have to compute the ##\phi(v_i)##. (Did I understand your definition of ##M^{B,\mathcal E}## correctly?)
     
  4. Jun 3, 2013 #3
    i don't think i have understood what you mean :(. could you please do an example of the calculation you have done by using a vector?
     
  5. Jun 3, 2013 #4

    Fredrik

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    I don't want to give away too much information. We only give hints here, not complete solutions. But I can of course explain what I did there. I just rewrote ##e_i## as a linear combination of the ##v_i## and then I used that ##\phi## is linear.

    The ##\phi(v_i)## are easy to calculate, since you've been given the matrix representation of ##\phi## in the ##\{v_i\}## basis.
     
  6. Jun 3, 2013 #5
    thank you :) solved it!
     
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