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Math Help - Linear transformations

  1. #1
    Junior Member SirOJ's Avatar
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    Linear transformations


    (b) Let
    L : R3 ->R2 be the linear transformation defined by


    L
    (x, y, z) = (x y, y z)

    (i) Find the standard (2
    3) matrix representation of L and compute L(1, 1, 1).

    I know how to show something is a linear transformation but unsure how to start this question. Just the starting point is all i'm looking for...
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Find the image of (1,0,0),(0,1,0),(0,0,1) under this transformation. The vectors you will obtain will be the columns of the matrix.
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  3. #3
    Junior Member SirOJ's Avatar
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    Quote Originally Posted by Bruno J. View Post
    Find the image of (1,0,0),(0,1,0),(0,0,1) under this transformation. The vectors you will obtain will be the columns of the matrix.
    I'm a little confused as to what you mean above.. Is there any way you could elaborate a small bit?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Sure!

    You have :

    L(1,0,0)=(1,0)

    L(0,1,0)=(-1,1)

    L(0,0,1)=(0,-1)

    and the three vectors on the right are the columns of the matrix which represents L with respect to the standard basis \beta. So we have

    [L]_\beta=\left(\begin{array}{ccc}1& -1& 0 \\ 0 &1 &-1 \\ \end{array}\right).
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  5. #5
    Math Engineering Student
    Krizalid's Avatar
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    it's worth to say that this procedure only applies when finding the associated matrix in the standard basis.
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  6. #6
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    But to generalize: If L is a linear transformation from vector space U, with dimension n, to vector space V, with dimension m, then it can be written in matrix form for given basis \{u_1, u_2, ..., u_n\} of U and given basis \{v_1, v_2, ..., v_m\} of V (changing bases will result in different but "equivalent" matrices representing the same transformation).

    Take L(u_1) and write it as a linear combination of the basis \{v_1, v_2, ...v_n\}, which you can do since L(u_1) is in V. Say, L(u_1)= a_1v_1+ a_2v_2+ ...+ a_mv_m. Then \begin{bmatrix}a_1 \\ a_2 \\ \cdot \\\cdot \\\cdot \\ a_m \end{bmatrix} is the first column in the matrix. Similarly, writing L{u_2} as a linear combination of v_1, v_2, ..., v_m gives the second column, etc.
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  7. #7
    Junior Member SirOJ's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    But to generalize: If L is a linear transformation from vector space U, with dimension n, to vector space V, with dimension m, then it can be written in matrix form for given basis \{u_1, u_2, ..., u_n\} of U and given basis \{v_1, v_2, ..., v_m\} of V (changing bases will result in different but "equivalent" matrices representing the same transformation).

    Take L(u_1) and write it as a linear combination of the basis \{v_1, v_2, ...v_n\}, which you can do since L(u_1) is in V. Say, L(u_1)= a_1v_1+ a_2v_2+ ...+ a_mv_m. Then \begin{bmatrix}a_1 \\ a_2 \\ \cdot \\\cdot \\\cdot \\ a_m \end{bmatrix} is the first column in the matrix. Similarly, writing L{u_2} as a linear combination of v_1, v_2, ..., v_m gives the second column, etc.

    I'm new to a lot of this stuff so some of that definition didn't really make sense to me... You are saying that i would have to approach a problem like the one below differently, correct?

    (b) Let
    L : R3 ->R2 be the linear transformation defined by

    L
    (x, y, z) = (x + y + z, y + z)

    (i) Find the standard (2
    3) matrix representation of L and compute L(0,1, 1).
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