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Help with basic linear algebra

  1. Jan 24, 2008 #1
    1. The problem statement, all variables and given/known data Theres two questions I need help with on m homework:

    I need to prove algebraically that the linear system r + 2s = -b1 and 3r+5s = b2 has a solution for all numbers b1, b2

    also: for vectors v and w prove that v-w and v+w are perpendicular if and only if the magnitude of v equals the magnitude of w.


    3. The attempt at a solution The first question i multiplied the first equation by -3 and added the two equations together to get 11s = b2 - 3b1 but i have no idea where to go from there.

    The second I proved that two vectors are perpendicular if their dot product is zero. I did the dot product of v-w and v+w an dgot [v1^2 - w1^2, . . . vn^2 - wn^2].here agian in stuck. any help please?
     
  2. jcsd
  3. Jan 24, 2008 #2

    Defennder

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    When you say prove algebraically, are you allowed to use matrices here? It's a lot easier if you could do so. Start by representing the linear system as an augmented matrix:

    [tex]\left(\begin{array}{*{20}c}1&2&-b_{1}\\3&5&b_{2}\end{array}\right)[/tex]

    If you want to show that there are exactly 1 solutions for both r,s , you need to show that you can reduce the augmented matrix (the sub-matrix on the left) above to the identity matrix.

    For the 2nd part, your approach is correct, but you should get this:

    [tex](v+w)\cdot(v-w) = v\cdot v + v\cdot w - w\cdot v - w\cdot w [/tex]

    You know where to go from here, right?
     
  4. Jan 24, 2008 #3
    You are almost done. Solve for s, and sustitude the result in one of original the equations in order to find r.

    Write the dot product [itex](\vec{v}-\vec{w})\cdot (\vec{v}+\vec{w})=0[/itex] and expand it.
     
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