Linear Algrebra- Orthogonal Vectors

[dondraper5]

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.

 

[HallsofIvy]

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

To show that take the dot product of $a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_mv_m$ with each of $v_1, v_2, v_3, \cdot\cdot\cdot, v_m$ in turn.

 

[dondraper5]

^ thank you, makes sense now.

 

[dondraper5]

any ideas for b?

 

[Fredrik]

any ideas for b?

Use the definitions of the $\|x\|$ notation on both sides. If you're allowed to use that the map $x\mapsto\|x\|$ 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:
$$\begin{align} &\|u+v\|^2=\cdots\\ &\|u-v\|^2=\cdots \end{align}$$ Think about the relationship between the norm and the inner product. Once you have used that, the rest will be easy.

 

[dondraper5]

^ thanks a lot

 

[I like Serena]

Umm... (b) is not true...

 

[Fredrik]

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