Let $\langle , \rangle$ : $\mathbb{R} ^2\times\mathbb{R} ^2\rightarrow \mathbb{R}$ be an inner product with the following properties:
$\langle (1,0) ,(1,0) \rangle=4$
$\langle (1,1) ,(1,1) \rangle=1$
$\langle (1,0) ,(3,3) \rangle=0$
I need to prove $B=\{(0,\frac{1}{\sqrt{5}}),\frac{1}{\sqrt{20}}(5,4)$} is an orthonormal basis of $\mathbb{R} ^2$. It seems simple but i dont know where to start. Any help would be appreciated. I am not asking for an answer just for a tip or something.
The vectors $(1,0)$ and $(1,1)$ form a basis for $\mathbb R^2$ and you’re given all of the values of the inner products of these two vectors with themselves and each other. As inner products are bilinear, that gives you enough information to compute the inner product of any two vectors. You’ll need to find the coordinates of $\left(0,1/\sqrt5\right)$ and $\left(1/\sqrt{20}(5,4)\right)$ with respect to this basis.