Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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
    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

    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook