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Lin algebra: find the matrix with respect to basis

  1. Dec 9, 2006 #1
    1. The problem statement, all variables and given/known data

    Let W be a 4dim vector space with basis {e1, e2, e3, e4}. Let T be the linear mapping:

    T(e1) = -e1 -2e2 + 2e3
    T(e2) = 4e1 + 4e2 - 5e3 -3e4
    T(e3) = 2e1 + 2e2 -3e3 -2e4
    T(e4) = -e2 + e3

    Let V be the subspace spanned by {e1 + e2 - e3, e1 - e4, -e1 + e2 -e3 +2e4}

    Now: find a basis for V and calculate the matrix T with respect to V (the matrix T restricted to the subspace V) with respect to this basis

    2. Relevant equations

    -

    3. The attempt at a solution

    well the 3 elements in the span of V are lin. DEP and i found that {e1 + e2 - e3, e1 - e4} are lin ind so they form a basis for V.

    Now, for the matrix.. I keep getting confused. It seems that I am beginning in R4 and ending in R2... i am confused on how to get from a 4x4 matrix 2x4? I calcluated what the basis vectors for V look like when they go through the transformation T (** I got: T(e1 + e2 - e3) = e1 - e4 ; T(e1 - e4) = - e1 - e2 + e3).

    I was thinking I would multiply the matrix rep. of T by something to give me the matrix rep. of **. Is this correct for what I should be doing?

    But it doesnt really make sense...

    I know my final answer needs to be a square matrix because later parts of this exercise ask me to calculate the eigenvalues (so it must be square.)

    Hopefully I did not really confuse you. All the examples I have been looking at to try and help don't seem to be completely relevant.
     
  2. jcsd
  3. Dec 9, 2006 #2

    matt grime

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    You've found a basis of V. Now, does T map V into itself? If so it is an linear map on T. What is it?
     
  4. Dec 9, 2006 #3
    since T is linear,
    T(V) =
    T(e1 - e4) = T(e1) - T(e4) = -e1 -e2 + e3
    T(e1 + e2 - e3) = T(e1) -T(e2) + T(e3) = e1 - e4


    the results come out of elements of the span, or linear combinations of elements of the span of V.

    therefore, T(V) is contained in V, so T maps V into itself..
     
  5. Dec 9, 2006 #4

    matt grime

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    So if we let f_1 and f_2 stand for the basis vectors e_1 -e_4 and e_1+e_2-e_3. T sends f_1 to what (in terms of f_i)?
     
  6. Dec 9, 2006 #5
    ah okay. then:

    T(f_1) = - f_2
    and
    T(f_2) = f_1

    .i feel we are now ready to construct the matrix knowing this but i am not sure what it is yet. im thinking
     
  7. Dec 9, 2006 #6
    would then matrix then just be:

    0 -1
    1 0

    ?
     
  8. Dec 9, 2006 #7

    matt grime

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    You shouldn't need to ask; it is a matter of verifying if it does what it is supposed to. It is the standard 'plugging the answer back in' to check.
     
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