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Modules with multiple operators

  1. Jun 4, 2005 #1
    Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition.

    [itex]\mathbb{R}^3[/itex] is module defined over this ring.

    So, we have three dimensional vectors whose elements are 2x2 matrices.

    My question: Can I also define another "scalar multiplication" that is over the field of real numbers (well, I know you can)...what is such a structure called? For example, I want it to do the following:

    [tex]
    3
    \begin{pmatrix}
    \begin{pmatrix}
    a & b\\
    c & d
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 1\\
    1 & 1
    \end{pmatrix}\\
    \begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 2\\
    4 & 3
    \end{pmatrix}
    \end{pmatrix}
    =
    \begin{pmatrix}
    3
    \begin{pmatrix}
    a & b\\
    c & d
    \end{pmatrix}
    &
    3\begin{pmatrix}
    1 & 1\\
    1 & 1
    \end{pmatrix}\\
    3\begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    3\begin{pmatrix}
    1 & 2\\
    4 & 3
    \end{pmatrix}
    \end{pmatrix}
    =
    \begin{pmatrix}
    \begin{pmatrix}
    3a & 3b\\
    3c & 3d
    \end{pmatrix}
    &
    \begin{pmatrix}
    3 & 3\\
    3 & 3
    \end{pmatrix}\\
    \begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    \begin{pmatrix}
    3 & 6\\
    12 & 9
    \end{pmatrix}
    \end{pmatrix}
    [/tex]

    in addition to:

    [tex]
    \begin{pmatrix}
    1 & 0\\
    0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    \begin{pmatrix}
    a & b\\
    c & d
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 1\\
    1 & 1
    \end{pmatrix}\\
    \begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 2\\
    4 & 3
    \end{pmatrix}
    \end{pmatrix}
    =
    \begin{pmatrix}
    \begin{pmatrix}
    1 & 0\\
    0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    a & b\\
    c & d
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 0\\
    0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    1 & 1\\
    1 & 1
    \end{pmatrix}\\
    \begin{pmatrix}
    1 & 0\\
    0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 0\\
    0 & 1
    \end{pmatrix}
    \begin{pmatrix}
    1 & 2\\
    4 & 3
    \end{pmatrix}
    \end{pmatrix}
    =\begin{pmatrix}
    \begin{pmatrix}
    a & b\\
    c & d
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 1\\
    1 & 1
    \end{pmatrix}\\
    \begin{pmatrix}
    0 & 0\\
    0 & 0
    \end{pmatrix}
    &
    \begin{pmatrix}
    1 & 2\\
    4 & 3
    \end{pmatrix}
    \end{pmatrix}
    [/tex]
     
  2. jcsd
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