From 1ac1357b767633cd7122be0ffd5e09068e7fdfc4 Mon Sep 17 00:00:00 2001 From: InkSoul Date: Sat, 5 Aug 2023 23:24:55 +0800 Subject: [PATCH] =?UTF-8?q?=E5=AE=8C=E6=88=90=E3=80=8A=E7=BA=BF=E6=80=A7?= =?UTF-8?q?=E4=BB=A3=E6=95=B0=E3=80=8B=E7=9F=A9=E9=98=B5=E4=B8=8E=E7=BA=BF?= =?UTF-8?q?=E6=80=A7=E5=8F=98=E6=8D=A2?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/about/_index.md | 2 +- .../mathematics/《线性代数》矩阵与线性变换.md | 121 ++++++++++++++++++ .../数学/《线性代数》矩阵与线性变换/剪切.png | Bin 0 -> 379438 bytes .../矩阵乘法与线性变换的关系.png | Bin 0 -> 233638 bytes .../《线性代数》矩阵与线性变换/线性变换推论.png | Bin 0 -> 208872 bytes 5 files changed, 122 insertions(+), 1 deletion(-) create mode 100644 content/mathematics/《线性代数》矩阵与线性变换.md create mode 100644 static/images/数学/《线性代数》矩阵与线性变换/剪切.png create mode 100644 static/images/数学/《线性代数》矩阵与线性变换/矩阵乘法与线性变换的关系.png create mode 100644 static/images/数学/《线性代数》矩阵与线性变换/线性变换推论.png diff --git a/content/about/_index.md b/content/about/_index.md index 96c9b3e..a23d569 100644 --- a/content/about/_index.md +++ b/content/about/_index.md @@ -32,7 +32,7 @@ date: 2022-05-10T17:23:35+08:00 这里存放一些我在学习高数,线代,概率论的一些随笔,一方面是图形学的琢磨用的比较多,另一方面也是考研做题需求 -线性代数部分基于[3Blue1Brown](https://space.bilibili.com/88461692)制作的[线性代数的本质](https://www.bilibili.com/video/BV1ys411472E/),如有错误欢迎页脚邮件指出 +线性代数部分基于[3Blue1Brown](https://space.bilibili.com/88461692)制作的[线性代数的本质](https://www.bilibili.com/video/BV1ys411472E),如有错误欢迎页脚邮件指出 ------------------------------------- diff --git a/content/mathematics/《线性代数》矩阵与线性变换.md b/content/mathematics/《线性代数》矩阵与线性变换.md new file mode 100644 index 0000000..405e066 --- /dev/null +++ b/content/mathematics/《线性代数》矩阵与线性变换.md @@ -0,0 +1,121 @@ +--- +title: "《线性代数》矩阵与线性变换" +date: 2023-08-05T22:33:01+08:00 + +--- + +## 线性变换 + +本质是将输入向量转换为输出向量的函数 + +线性有两点性质 +1. 直线在变换后仍然保持为直线,不能产生弯曲 +2. 原点保持固定 + +线性变换可被视为保持网格线平行且等距分布的变换 + +为了使用数值描述线性变换只需要记录两个基向量$\hat{i}$$\hat{j}$变换后的位置,其他向量也会随之变换 + +由线性变换结论可知变换后向量位置是原向量位置与变换后向量相乘 + +![](../../images/数学/《线性代数》矩阵与线性变换/线性变换推论.png) + +由此我们可以推导出在未知变换方式情况下计算变换后向量的方法 + +假设 + +$\hat{i} = \begin{bmatrix} + 1 \\\\ + -2 +\end{bmatrix}$ + +$\hat{j} = \begin{bmatrix} + 3 \\\\ + 0 +\end{bmatrix}$ + +一般情况下,设一个向量的坐标为(x,y),可得变换后坐标如下 + +$$\begin{bmatrix} + x \\\\ + y +\end{bmatrix} +\rightarrow x +\begin{bmatrix} + 1 \\\\ + -2 +\end{bmatrix}+ +y +\begin{bmatrix} + 3 \\\\ + 0 +\end{bmatrix}= +\begin{bmatrix} + 1x+3y \\\\ + -2x+0y +\end{bmatrix} +$$ + +线性变换中存在一种特殊形式,《错切》(剪切) + +$\hat{i}$保持不变,$\hat{j}$移动到坐标(1,1) + +![](../../images/数学/《线性代数》矩阵与线性变换/剪切.png) + + +## 矩阵 + +由上部分可知,一个二维线性变换仅由四个数字完全确定,变换后的$\hat{i}$的两个坐标和变换后的$\hat{j}$的两个坐标,通常通常这些坐标会被包装在一个2$\times$2的格子中,即2$\times$2矩阵,第一列为变换后i的坐标,第二列为变换后j的坐标 + +对于一个矩阵和一个初始向量,要得知变换后的向量位置只需要将向量的坐标与矩阵的特定列分别相乘后相加即可 + +矩阵通常是记录线性变换信息的工具 + +一般情况下 + +已知$\begin{bmatrix} + a & b \\\\ + c & d +\end{bmatrix}$要得知$\begin{bmatrix} + x \\\\ + y +\end{bmatrix}$变换后的坐标,可有以下公式得到 + +$$ +\begin{bmatrix} + a & b \\\\ + c & d +\end{bmatrix} +\begin{bmatrix} + x \\\\ + y +\end{bmatrix} = +x +\begin{bmatrix} + a \\\\ + b +\end{bmatrix} + +y +\begin{bmatrix} + b \\\\ + d +\end{bmatrix} = +\begin{bmatrix} + ax+by \\\\ + cx + dy +\end{bmatrix} +$$ + +由此也可看作矩阵向量的乘法 + +在思考时,不妨将矩阵的列视为变换后的基向量,将矩阵向量乘法看作它们的线性组合更加便于数形结合加以理解 + +![](../../images/数学/《线性代数》矩阵与线性变换/矩阵乘法与线性变换的关系.png) + +由矩阵反向推演线性变换的过程: + +* 将$\hat{i}$和$\hat{j}$平移到列对应的坐标上,空间上剩余部分便于随二者移动,以保持网格线平行且等距分布 +* 如果两者是列线性相关的,则其中一个向量是另一个的倍数,空间会被挤压到它们所在的直线上,即它们组成的一维空间 + + +**矩阵都可以解读为对空间的一种特定变换** diff --git a/static/images/数学/《线性代数》矩阵与线性变换/剪切.png b/static/images/数学/《线性代数》矩阵与线性变换/剪切.png new file mode 100644 index 0000000000000000000000000000000000000000..52f27d62c2ac069c8d55c75f01ee7890db6d9aac GIT binary patch literal 379438 zcmXV1Wl&px*M!m*XdyTa5TLkQp+L~$?oNT?uED(&O0bgP#a)V9aVZ3MD^Og5yF0x6 zpLsuIGB-2H%AJnJ2@%Xs^9OyTC%m5Y{Yzq(=%LW92l zTTEU|{+GT}$Hp_Xe|U8kuyl0CCv@aZ{ejuZa&e8BNr78b$i3$92buF)SNPe>#k^hU z7io2Md>^9#k`Y>lSdil@t5^fG_oJ^XCWA{$K`&p3-5dZFugj$P~)J$|y@kqj!+M90Ke^B&Hm5(6e 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