Exercise 4.4.12

(a) Orthonormal $a$ ’s: $a_1^{T}b=a_1^T\left(x_1{a}_1+x_2{a}_2+x_3{a}_3\right)=x_1\left(a_1^T{a}_1\right)=x_1$

(b) Orthogonal $a$ ’s: $a_1^{T}b=a_1^T\left(x_1{a}_1+x_2{a}_2+x_3{a}_3\right)=x_1\left(a_1^T{a}_1\right)$. Therefore $x_1=a_1^{T}b∕a_1^T{a}_1$

(c) $x_1$ is the first component of $A^{-1}$ times $b$ ( $A$ is 3 by 3 and invertible).