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Homework Help: Need to rewrite linear combination as vector expression.

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Given vectors
    [itex]
    {\bf r}=\left[r_1,r_2,r_3\ldots{}r_n\right]^T
    [/itex]
    [itex]
    {\bf e}=\left[e_1,e_2,e_3\ldots{}e_n\right]^T
    [/itex]

    I need to write the sum

    [itex]
    \sum_{i=1}^{n}r_ie_i^2
    [/itex]

    in terms of [itex]{\bf r}[/itex] and [itex]{\bf e}[/itex]

    2. Relevant equations
    Nil.

    3. The attempt at a solution

    Without [itex]r_i[/itex], I am able to write [itex]\sum_{i=1}^{n}e_i^2[/itex] as

    [itex]
    \sum_{i=1}^{n}e_i^2={\bf e}^T{\bf e}
    [/itex]

    If [itex]r_i[/itex] can be independant of [itex]i[/itex] I should be able to move it out of the summation. I am looking for some expansion/reexpression of [itex]r_i[/itex].
     
  2. jcsd
  3. Feb 7, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi devonho! :smile:
    why?? :confused:

    (i don't think you can)
     
  4. Feb 14, 2012 #3
    This seems to work:

    [itex]
    {\bf r}=
    \left[
    \begin{array}{ccccc}
    r_1 & 0 & \ldots & & 0\\
    0 & r_2 & & & \vdots \\
    \vdots & & r_3 & & \\
    & & & \ddots & \\
    0 & \ldots & & & r_n \\
    \end{array}
    \right]
    [/itex]

    [itex]
    {\bf e}=
    \left[
    \begin{array}{c}
    e_1 \\
    e_2 \\
    \vdots\\
    e_n \\
    \end{array}
    \right]
    [/itex]

    [itex]
    {\bf e}^T{\bf re}=\left[ e_1, e_2 \ldots e_n\right]
    \left[
    \begin{array}{ccccc}
    r_1 & 0 & \ldots & & 0\\
    0 & r_2 & & & \vdots \\
    \vdots & & r_3 & & \\
    & & & \ddots & \\
    0 & \ldots & & & r_n \\
    \end{array}
    \right]
    \left[
    \begin{array}{c}
    e_1 \\
    e_2 \\
    \vdots\\
    e_n \\
    \end{array}
    \right]
    =r_1e_1^2+r_2e_2^2\ldots +r_ne_n^2
    =\sum_{i=1}^{n}r_ie_i^2
    [/itex]
     
  5. Feb 14, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    but devonho, how do you form that middle matrix out of the vector r ? :confused:
     
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