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# 矩阵变换

## 乘法作为变换

"变换" 的想法最初看起来可能会比真正的情况复杂得多，所以在开始讲解 $2×2$$2$-维空间的变换，或者 $3×3$ 矩阵在 $3$-维空间变换前，我们先看看简单的数字 (即 $1×1$ 的矩阵) 可以被看做 $1$-维空间上的变换。
"$1$-维空间"只是一个简单的数轴。

## 关注线性变换中的特定向量

$\left[\begin{array}{c}1\\ 1\end{array}\right]\to \left[\begin{array}{c}4\\ -2\end{array}\right]$

$\left[\begin{array}{c}2\\ 0\end{array}\right]\to 2\cdot \left[\begin{array}{c}1\\ -2\end{array}\right]=\left[\begin{array}{c}2\\ -4\end{array}\right]$.

$\begin{array}{rl}\left[\begin{array}{c}x\\ 0\end{array}\right]=x\cdot \left[\begin{array}{c}1\\ 0\end{array}\right]& \to x\cdot \left[\begin{array}{c}1\\ -2\end{array}\right]=\left[\begin{array}{c}x\\ -2x\end{array}\right]\end{array}$

$-1\cdot \left[\begin{array}{c}1\\ -2\end{array}\right]+2\cdot \left[\begin{array}{c}3\\ 0\end{array}\right]=\left[\begin{array}{c}5\\ 2\end{array}\right]$

## 用矩阵表示二维线性变换

$\left[\begin{array}{c}x\\ y\end{array}\right]=x\left[\begin{array}{c}1\\ 0\end{array}\right]+y\left[\begin{array}{c}0\\ 1\end{array}\right]$

$x\cdot \left[\begin{array}{c}a\\ c\end{array}\right]+y\cdot \left[\begin{array}{c}b\\ d\end{array}\right]=\left[\begin{array}{c}ax+by\\ cx+dy\end{array}\right]$

$\mathbf{\text{A}}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$\mathbf{\text{Av}}=\left[\begin{array}{c}ax+by\\ cx+dy\end{array}\right]$