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Linear transformation help needed

  1. Jun 18, 2009 #1
    1. The problem statement, all variables and given/known data

    let T : R3 --> R2 be the transformation defined by T([x]) = [y+z]
    y x+z
    z

    (a) show that T is a linear transformation

    (b) calculate [T]B',B- the matrix of T with respect to the bases B and B' where
    1 0 0
    B = { [0] [1] [0] } and B' = {[1] [-1]}
    0 0 1 1 1
    1
    (c) determine the coordinates of T{[1]} with respect to basis B'
    1
    2. Relevant equations

    nothing much can help...

    3. The attempt at a solution

    part (a) is fine, can be proved

    but have little problem in part (b)

    since B is 3*3 matrix and B' is 2*2, i can't keep going on that

    i've got [T]B={[0] [1][1]}
    [1] [0][1]

    and i can't keep going from here

    and same as part (c) , how can i transfer a 3*1 into with respect 2*2
     
  2. jcsd
  3. Jun 18, 2009 #2
    i don't know whats wrong with my typing

    i tis not showing right

    but B is standard basis for R3 100,010,001

    and B' is [1 -1; 1 1]
     
  4. Jun 18, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Use LaTex! Web sites in general do not respect spaces.

    The linear transformation, T, is given by
    [tex]T\left(\begin{bmatrix}x \\ y \\ z\end{bmatrix}\right)= \begin{bmatrix} y+ z \\ x+ z\end{bmatrix}[/tex]
    (Click on the formula to see the code.)

    To show that this is a linear transformation, you must show that T(u+ v)= T(u)+ T(v) for any u, v in R3 and that T(au)= aT(u) for any u in R3 and any real number a.

    If T:U-> V and you are given bases for U and V, to write T as a matrix do this:
    Apply T to each of the basis vector of U in turn. Write the result of each as a linear combination of the basis vector for V. The coefficients of each linear combination is a column in the matrix. For example
    [tex]T\left(\begin{bmatrix}0 \\ 0 \\ 1\end{bmatrix}\right)= \begin{bmatrix}1 \\ 1\end{bmatrix}= 0\begin{bmatrix}1 \\ -1\end{bmatrix}+ 1\begin{bmatrix}1 \\ 1\end{bmatrix}[/tex]
    so the third column of the matrix is
    [tex]\begin{bmatrix}0 \\ 1\end{bmatrix}[/tex]

    Note that the order in which the basis vectors are given is important. [0 0 1] is the third vector in the basis for R3 so applying T to it gives the third column of the matrix.
     
    Last edited: Jun 18, 2009
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