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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 $[id]_\alpha^\beta = ([id]_\beta^\alpha)^{-1}$.

          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 $[id]_\alpha^\beta = ([id]_\beta^\alpha)^{-1}$.

$Problem 87 Let id: ℝ^n →ℝ^n be the identity transformation. Let xk denote the vector in the n-space ℝ^n whose first k - 1 entries are zero and the last n - k + 1 entries are 1. Then β = {x1, x2, …, xn} is a basis for ℝ^n (see Problem 60). Let α = {e1, e2, …, en} be the standard (ordered) basis for ℝ^n. Show that [id]β = [x1 x2 … xn]. Try to find the change of basis matrix [id]α^β. Which is easier to find, [id]α^β or [id]β^α? Hint: It is easy to find the change of basis matrix [id]β^α if α is the standard (ordered) basis for ℝ^n and [id]α^β = ([id]β^α)^-1.$

Added by Bobby W.

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