Answer
The set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$is orthogonal, not orthonormal and it is a basis for $R^2$.
Work Step by Step
given $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$
let $v_1=(1,2),v_2=(-\frac{2}{5},\frac{1}{5})$
(a)
since $v_1v_2=-\frac{2}{5}+\frac{2}{5}=0$
then, the set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is orthogonal.
(b)
scince
$\left\|{v}_{1}\right\|=\sqrt{v_{1} \cdot v_{1}}=\sqrt{1+4}=\sqrt{13}\neq1$
then, the set $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is not orthonormal.
(c)
by the corollary to Theorem 5.10, $\{(1,2),(-\frac{2}{5},\frac{1}{5})\}$ is a basis for $R^2$.
