## 每週問題 June 13, 2016

Let $\boldsymbol{\beta}=\{\mathbf{x}_1,\ldots,\mathbf{x}_k\}$ and $\boldsymbol{\gamma}=\{\mathbf{y}_1,\ldots,\mathbf{y}_k\}$ be bases for a subspace $\mathcal{V}$ in $\mathbb{R}^n$. Let $X=\begin{bmatrix} \mathbf{x}_1&\cdots&\mathbf{x}_k \end{bmatrix}$ and $Y=\begin{bmatrix} \mathbf{y}_1&\cdots&\mathbf{y}_k \end{bmatrix}$. Show that the change of coordinates matrix from $\boldsymbol{\beta}$ to $\boldsymbol{\gamma}$ is

$P=(Y^TY)^{-1}Y^TX$.

$[\mathbf{v}]_{\boldsymbol{\beta}}=\begin{bmatrix} c_1\\ \vdots\\ c_k \end{bmatrix},~~[\mathbf{v}]_{\boldsymbol{\gamma}}=\begin{bmatrix} d_1\\ \vdots\\ d_k \end{bmatrix}$

$[\mathbf{v}]_{\boldsymbol{\gamma}}=(Y^TY)^{-1}Y^TX[\mathbf{v}]_{\boldsymbol{\beta}}$