Exercise 18

Theorem 2.7 describes a change of basis from any basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{k}\rangle$ to one that is orthogonal $K=\langle\vec{\kappa}_{1},\ldots,\vec{\kappa}_{k}\rangle$. Consider the change of basis matrix ${\rm Rep}_{B,K}(\operatorname{id})$.

• Prove that the matrix ${\rm Rep}_{K,B}(\operatorname{id})$ changing bases in the direction opposite to that of the theorem has an upper triangular shape — all of its entries below the main diagonal are zeros.

• Prove that the inverse of an upper triangular matrix is also upper triangular (if the matrix is invertible, that is). This shows that the matrix ${\rm Rep}_{B,K}(\operatorname{id})$ changing bases in the direction described in the theorem is upper triangular.