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Scalar multiplication(vectors)

  1. Jan 24, 2005 #1
    Hi everybody,
    I have a small question. I know that we have defined multiplication of a number and a vector ,for example b*A (capital letters =vectors, everything else=real numbers). We have also defined that b*(c*A)=(b*c)*A. From these two rules is a*b*c*d*...*k*Z defined (= product of n numbers with a vector) without using parentheses? What about a*b*c*E*D ? And one last thing: is scalar multiplication also written with juxtaposition? For example the above examples can be written like this: abcd...kZ and abcE*D ?
    Thanks
     
  2. jcsd
  3. Jan 25, 2005 #2

    cronxeh

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    Gold Member

    You can multiply a scalar by a matrix anytime you want. However in order to multiply a matrix by another matrix their size has to be compatible.

    [tex] A_{m x n} * B_{n x p} [/tex]. For example,
    [tex]A _ {2 x 3} = \left(
    \begin{array}{x1x2x3}
    2 & 4 & 3\\
    1 & -1 & 5\\
    \end{array}
    \right)
    [/tex] can only be multiplied by a matrix which is in [tex]B_{3 x n}[/tex] form.

    So let [tex] B_{3x5}= \left(
    \begin{array}{x1x2x3x4x5}
    1 & 0 & 5 & 2 & 3\\
    0 & -1 & 2 & 4 & 1\\
    4 & 5 & 6 & 7 & 8
    \end{array}
    \right)
    [/tex]

    The resulting matrix will be [tex]A_{2x3} * B_{3x5} = C_{2x5}[/tex]
    [tex]C_{2x5} = \left(
    \begin{array}{x1x2x3x4x5}
    14 & 11 & 36 & 41 & 34\\
    21 & 26 & 33 & 33 & 42\\
    \end{array}
    \right)[/tex]

    If you have [tex](a*b*c*d) * (A_{3x5} * B_{5x4} * C_{4x7}[/tex] The resulting matrix will be: [tex](a*b*c*d) * (M_{3x7})[/tex]
     
  4. Jan 25, 2005 #3

    jcsd

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    Science Advisor
    Gold Member

    a*b*c*E*D is not defined without first defining the multiplication of two vectors (so in otherwords you'd have to say exactly what E*D means).

    It is usual to use juxtapostion for the multiplication of two scalars or the multiplication of a scalar by a vector. As there is more than one kind of product of two vectors, it's usual to use whatever binary operator denotes that product to avoid confusion.
     
  5. Jan 26, 2005 #4
    Thanks for your answers
     
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