Problem 87 Let $id: \mathbb{R}^n \to \mathbb{R}^n$ be the identity transformation. Let $x_k$ denote the vector in the $n$-space $\mathbb{R}^n$ whose first $k - 1$ entries are zero and the last $n - k + 1$ entries are 1. Then $\beta = \{x_1, x_2, \dots, x_n\}$ is a basis for $\mathbb{R}^n$ (see Problem 60). Let $\alpha = \{e_1, e_2, \dots, e_n\}$ be the standard (ordered) basis for $\mathbb{R}^n$. Show that $[id]_\beta = [x_1 \ x_2 \dots x_n]$. Try to find the change of basis matrix $[id]_\alpha^\beta$. Which is easier to find, $[id]_\alpha^\beta$ or $[id]_\beta^\alpha$? Hint: It is easy to find the change of basis matrix $[id]_\beta^\alpha$ if $\alpha$ is the standard (ordered) basis for $\mathbb{R}^n$ and $[id]_\alpha^\beta = ([id]_\beta^\alpha)^{-1}$.
