Show that $\{\mathbf{u},\mathbf{v}\}$ is linearly independent.

Question

Let $n\in \mathbb{N}$ and let $\mathbf{u},\mathbf{v}$ be nonzero vectors in $\mathbb{R}^n$ such that $\mathbf{u}\cdot \mathbf{v} = 0$. Show that $\{\mathbf{u},\mathbf{v}\}$ is linearly independent.
Just looking for a place to start thank you!

Answer 1

Let $au+bv=0$, where $\{a,b\}\subset\mathbb R$.

Thus, $$(au+bv)\cdot u=0$$ or

$$a(u\cdot u)+b(v\cdot u)=0$$ or $$a(u\cdot u)=0.$$ Can you end it now?

Answer 2

Note that in the case of only two vectors, linear dependence for nonzero vectors just means

$u=\alpha\,v\quad$ with $\alpha\neq 0$, in which case $u\cdot v=\alpha||v||^2\neq 0$.

The contraposition gives $u\cdot v=0\implies\{u,v\}$ linearly independent.