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Linear Algrebra- Orthogonal Vectors

  1. Nov 26, 2011 #1
    I am having trouble with these questions-

    Explain/prove whether:
    (a) Any set {v1,v2,.....vk} of orthogonal vectors in Rn is linearly independent.
    (b) If there is a vector v in Rn and scalar c in R, we have ||cv|| = c||v||
    (c) for any vectors u, v in Rn, ||u+v||^2 + ||u-v||^2 = 2 ||u||^2 + 2||v||^2

    I think part a is true, but can't get around a way to prove it. Need help with b and c.
     
    Last edited: Nov 26, 2011
  2. jcsd
  3. Nov 26, 2011 #2

    HallsofIvy

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    To show that vectors, [itex]\{v_1, v_2, v_3, \cdot\cdot\cdot, v_m\}[/itex] are independent, you must show that if [itex]a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_mv_m[/itex] then [itex]a_1= a_2= a_3= \cdot\cdot\cdot= a_m= 0[/itex].

    To show that take the dot product of [itex]a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_mv_m[/itex] with each of [itex]v_1, v_2, v_3, \cdot\cdot\cdot, v_m[/itex] in turn.
     
  4. Nov 28, 2011 #3
    ^ thank you, makes sense now.
     
  5. Nov 28, 2011 #4
    any ideas for b?
     
  6. Nov 28, 2011 #5

    Fredrik

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    Use the definitions of the [itex]\|x\|[/itex] notation on both sides. If you're allowed to use that the map [itex]x\mapsto\|x\|[/itex] is a norm, you can just compare the equality you've been given with the properties of a norm. I recommend you do it both ways. (They're both very easy).

    For c, you should look at the terms on the left, one at at a time:
    [tex]
    \begin{align}
    &\|u+v\|^2=\cdots\\
    &\|u-v\|^2=\cdots
    \end{align}
    [/tex] Think about the relationship between the norm and the inner product. Once you have used that, the rest will be easy.
     
    Last edited: Nov 28, 2011
  7. Nov 28, 2011 #6
    ^ thanks a lot
     
  8. Nov 28, 2011 #7

    I like Serena

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    Umm... (b) is not true...
     
  9. Nov 28, 2011 #8

    Fredrik

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    Right, but he said "explain/prove whether...", so he probably wasn't assuming that they were all true.
     
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