TITLE: Basic Linear Algebra Proof - Orthogonal Vectors
QUESTION [11 upvotes]: Prove that if $\mathbf{u}$ and $\mathbf{v}$ are nonzero orthogonal vectors in $\Bbb R^n$ they are linearly Independent.
I've struggled with this a bit, here is what I know so far:
Suppose $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.  Then $\mathbf{u\cdot v}=0$
and
$c_1\mathbf{u}+c_2\mathbf{v}=0$ is linearly Independent iff $c_1=c_2=0$
I know I need to end with $c_1=c_2=0$ but I can't find a path that reaches this conclusion.  I feel like I need to use properties of the dot product to connect my first assumption to my second assumption but I'm lost on the way.

REPLY [7 votes]: Just use contradiction. If $u,v$ are linearly dependent and non-zero, then $u = cv$ (note: $c$ is a function of your $c_1,c_2$). Then $u\cdot v \neq 0 \rightarrow$ contradiction.