let $a= (x_1,x_2)$ and $b=(y_1,y_2)$ be vectors in $\mathbb{R}^2$ such that $$x_1y_1 +x_2y_2=0$$ , $$x_1^2+ x_2^2 =y_1^2 +y_2^2 =1$$
It is given that $B=\{a, b\}$ is basis for $\mathbb{R}^2 $
find the coordinates of the vectors $(u,v)$ in the ordered basis $B=\{a,b\}$
My attempt : I take $(u,v)= c_1 a + c_2b= (c_1x_1 +c_2y_1 , c_1x_2 +c_2y_2)$
so the coordinates are $c_1x_1 +c_2y_1=u$ and $v= c_1x_2 +c_2y_2$
Best Answer
That basis is an orthnormal basis, and therefore the coordinates of $(u,v)$ with respect to that basis are $\bigl\langle(u,v),a\bigr\rangle(=ux_1+vx_2)$ and $\bigl\langle(u,v),b\bigr\rangle(=uy_1+vy_2)$. That is$$(u,v)=(ux_1+vx_2)a+(uy_1+vy_2)b.$$