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Matrix of linear transformation

  1. Apr 21, 2005 #1
    Hi

    I got a question regarding the matrix of linear transformation.

    A linear transformation L which maps [itex]\mathbb{R}^{3} \rightarrow \mathbb{R}^2[/itex] implies that L(2,-1,-1) = (0,0) and L(-1,2,1) = (1,3) and L(2,2,1) = (4,9).

    My question is: The matrix of linear transformation is that then?

    [itex]
    \left[\begin{array}{ccc}
    0 & 1 & 4\\
    0 & 3 & 9\\
    \end{array}\right]
    [/itex]


    Sincerely
    Fred
     
  2. jcsd
  3. Apr 21, 2005 #2
    Well, you may to remember that the matrix of a homomorphism or linear application, or transformation between to vectorial spaces is defined by the results of applying trasformation to basis vectors of any basis of the origin space (in this case a basis of [tex]\mathbb{R}^{3}[/tex]).

    This is not acompplished by the vectors that you give (they are NOT lineary independent). The most simple form to obtain a matrix of the transformation is using the property of linearity, it let you to make some linear combinations between the functions that you are given to get the vectors of the canonical basis and their transforms under the linear transformation.
     
  4. Apr 25, 2005 #3
    Hi and thanks for Your answer.

    I know that the definition of linear transformation[itex]\mathrm{L}:\mathbb{R}^m \rightarrow \mathbb{R}^n[/itex] is a follows.

    [itex]L(u+v) = L(u) + L(v) \ \ \mathrm{and} \ \ \mathrm{L(\alpha u)} = \alpha \mathrm{L(u)}[/itex]

    Do I then apply this definition to the given pre-conditions in my first post?

    In order to obtain the matrix of linear transformation?

    Sincerely

    Fred

    p.s.

    Is this matrix of linear transformation then ??
    [itex]
    \mathrm{L} \left(\begin{array}{ccc}
    2\\
    -1\\
    -1\\
    \end{array}\right) +
    \mathrm{L} \left(\begin{array}{ccc}
    -1\\
    2\\
    1\\
    \end{array}\right)
    =
    \left(\begin{array}{cc}
    1\\
    3
    \end{array}\right)
    [/itex]

    [itex]
    \mathrm{L} \left(\begin{array}{ccc}
    -1\\
    2\\
    1\\
    \end{array}\right) +
    \mathrm{L} \left(\begin{array}{ccc}
    2\\
    2\\
    1\\
    \end{array}\right)
    =
    \left(\begin{array}{cc}
    5\\
    12
    \end{array}\right)
    [/itex]

    The matrix A of linear transformation then being:

    [itex]
    A= \left [\begin{array}{cc}
    1 & 5 \\
    3 & 12
    \end{array}\right]
    [/itex]
     
    Last edited: Apr 25, 2005
  5. Apr 25, 2005 #4

    xanthym

    User Avatar
    Science Advisor

    SOLUTION HINTS:
    To obtain the Linear Transformation "L" for which:
    a) L(2,-1,-1) = (0,0)
    b) L(-1,2,1) = (1,3)
    c) L(2,2,1) = (4,9)
    the following simultaneous system of 6 equations in 6 unknowns (grouped to correspond to conditions "a", "b", and "c" above) must be solved:

    [tex] 1: \ \ \ \
    \begin{array}{rrrrr}
    \hline
    (2)L_{1,1} \ + & (-1)L_{1,2} \ + & (-1)L_{1,3} & = & 0 \\
    (2)L_{2,1} \ + & (-1)L_{2,2} \ + & (-1)L_{2,3} & = & 0 \\
    \hline
    (-1)L_{1,1} \ + & (2)L_{1,2} \ + & (1)L_{1,3} & = & 1 \\
    (-1)L_{2,1} \ + & (2)L_{2,2} \ + & (1)L_{2,3} & = & 3 \\
    \hline
    (2)L_{1,1} \ + & (2)L_{1,2} \ + & (1)L_{1,3} & = & 4 \\
    (2)L_{2,1} \ + & (2)L_{2,2} \ + & (1)L_{2,3} & = & 9 \\
    \hline
    \end{array}
    [/tex]

    The Linear Transformation "L" would then be represented by the following matrix:

    [tex] 2: \ \ \ \ \ \ \mathsf{L} \ \ = \ \ \left [
    \begin{array}{ccc}
    L_{1,1} & L_{1,2} & L_{1,3} \\
    L_{2,1} & L_{2,2} & L_{2,3} \\
    \end{array} \right ]
    [/tex]

    Equation System #1 above consists of 6 equations in 6 unknowns. The first step towards its solution is formation of the 6x6 coefficient matrix "M":

    [tex] 3: \ \ \ \ \ M \ \ = \ \ \left [
    \begin{array}{rrrrrr}
    2 & -1 & -1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 2 & -1 & -1 \\
    -1 & 2 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & -1 & 2 & 1 \\
    2 & 2 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 2 & 2 & 1 \\
    \end{array} \right ]
    [/tex]

    such that:

    [tex] 4: \ \ \ \ \ \ \ \ \left [
    \begin{array}{rrrrrr}
    2 & -1 & -1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 2 & -1 & -1 \\
    -1 & 2 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & -1 & 2 & 1 \\
    2 & 2 & 1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 2 & 2 & 1 \\
    \end{array} \right ]
    \left [
    \begin{array}{c}
    L_{1,1} \\
    L_{1,2} \\
    L_{1,3} \\
    L_{2,1} \\
    L_{2,2} \\
    L_{2,3} \\
    \end{array} \right ] \ \ = \ \ \left [
    \begin{array}{c}
    0 \\
    0 \\
    1 \\
    3 \\
    4 \\
    9 \\
    \end{array} \right ]
    [/tex]

    Solve for "L" by determining M(-1).
    (Hint #1: The latter inverse exists if det(M) ≠ 0).
    (Hint #2: det(M) = -9)
    (Hint #3: L = [ 1 0 2 ; 2 1 3 ] )



    ~~
     
    Last edited: Apr 25, 2005
  6. Apr 25, 2005 #5

    Hello and many thanks for your answer :smile:

    Please correct me if I understand You incorrectly.

    By determining the inverse matrix of M does that then give me the matrix of Linear transformation in [itex]\mathbb{R}^2[/itex] ??

    Sincerely Fred
    p.s. Again many thanks for Your answer.
     
  7. Apr 25, 2005 #6

    xanthym

    User Avatar
    Science Advisor

    The inverse M(-1) of matrix "M" is NOT itself the matrix representing Linear Transformation "L". Rather, the inverse M(-1) enables computation of elements Lj,k of the matrix "L" representing Linear Transformation "L". This is accomplished by solving the matrix equation:

    [tex] 5: \ \ \ \ M \cdot
    \left [
    \begin{array}{c}
    L_{1,1} \\
    L_{1,2} \\
    L_{1,3} \\
    L_{2,1} \\
    L_{2,2} \\
    L_{2,3} \\
    \end{array} \right ] \ \ = \ \ \left [
    \begin{array}{c}
    0 \\
    0 \\
    1 \\
    3 \\
    4 \\
    9 \\
    \end{array} \right ]
    [/tex]

    so that elements Lj,k are determined from:

    [tex] 6: \ \ \ \ \
    \left [
    \begin{array}{c}
    L_{1,1} \\
    L_{1,2} \\
    L_{1,3} \\
    L_{2,1} \\
    L_{2,2} \\
    L_{2,3} \\
    \end{array} \right ] \ \ = \ \ M^{-1} \cdot \left [
    \begin{array}{c}
    0 \\
    0 \\
    1 \\
    3 \\
    4 \\
    9 \\
    \end{array} \right ]
    [/tex]

    The Linear Transformation matrix "L" is then given by:

    [tex] 7: \ \ \ \ \ \ \mathsf{L} \ \ = \ \ \left [
    \begin{array}{ccc}
    L_{1,1} & L_{1,2} & L_{1,3} \\
    L_{2,1} & L_{2,2} & L_{2,3} \\
    \end{array} \right ]
    [/tex]

    Or in this case:

    [tex] \color{red} 8: \ \ \ \ \ \mathsf{L} \ \ = \ \ \left [
    \begin{array}{ccc}
    1 & 0 & 2 \\
    2 & 1 & 3 \\
    \end{array} \right ]
    [/tex]


    ~~
     
    Last edited: Apr 25, 2005
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