Finding orthogonal of a set in functional analysis

Question

I am self studying functional analysis from Kreyszig Functional analysis and I need help in Ex 3.3 Question no. 5 which is ------> If X = $ R^2 $ . Then find orthogonal complement of M if M is {x} , where x= (e1, e2) and x is non zero.

Please give some hints.

Answer 1

So $M^{\perp}$ is the set of all $y=(f_1,f_2)\in \mathbb{R}^2$ such that $\langle x,y\rangle=0$. Now, the inner product on $\mathbb{R}^2$ is given by

$$ \langle x,y\rangle=e_1f_1+e_2f_2, $$ which is $0$ when? This amounts to solving an equation.