$($a$)$ Using the basis $\{1,t,t^2\},$ defined for $t$ values from $-1$ to $+1,$ find the best mean$-$square approximation of the function shown in Figure $1($i$.$e$.$ minimize the Euclidean norm of the difference between this function and its approximation$).$ Is this basis orthogonal?

$($b$)$ Find the best linear mean$-$square approximation of the function in Figure $1$ using the basis $\{1,t\}.$ Is this basis orthogonal?

$($c$)$ Compare the errors of the two approximations in light of the fact that the second basis is a subset of the first.