Linear Algrebra- Orthogonal Vectors

  • Thread starter dondraper5
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  • #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.
 
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Answers and Replies

  • #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.
 
  • #3
^ thank you, makes sense now.
 
  • #4
any ideas for b?
 
  • #5
Fredrik
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any ideas for b?
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.
 
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  • #6
^ thanks a lot
 
  • #7
I like Serena
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Umm... (b) is not true...
 
  • #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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