1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Need to rewrite linear combination as vector expression.
Loading...