线性代数的本质是3b1b在bilibili上传的系列视频，通过python绘图将线性代数的矩阵计算可视化展现，并用“变换”这一概念理解线性代数。对于图形学和技术美术而言，理解线性代数的几何直观是非常重要的。
一 什么是向量
1.1 三种视角
- 数学-日常向量的表现形式
- 计算机-二元数组的表现形式-上为x轴，下为y轴
- 物理专业-带有长度和方向的矢量
1.2 概念理解
一个数组如何理解：例如
：在平面直角坐标系中，其含义就是从原点出发，顺x轴（第一个轴）移动a个单位，再顺y轴（第二个轴）移动b个单位。关于这两种说法的差别，要回到高中时期的田媛小课堂，i^ 和 j^ 的问题。（读作i hat 和 j hat,在后文中用i，j表示）所谓的i和j，本质上就是空间/平面中的坐标轴，在这两条坐标轴确定的空间中构建其他的向量和函数。
向量加法的图像合理性：将向量看作某种特定的运动，一个点从零点沿着其中一个向量移动到终点，再沿第二个向量移动。这是数轴加法的拓展。相对而言，数组形式的向量加法就是
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向量乘法的图像特征：由于向量是带有方向和大小的量，因此带有负数标量（Scalars）的乘法会将之反向。几何上称之为“缩放”/“Scaling”。
二 线性相关、张成空间与基
2.1 向量坐标的另一个视角
2.2 基向量
i 和 j 是xy坐标系的“基向量”，他们合起来称为坐标系的基。也就是说，当我们将坐标看作标量时，基向量就是这些标量缩放的对象。当我们使用数字描述向量时，它都依赖于我们正在使用的基。
2.3 线性组合
两个数乘向量的和被成为这两个向量的线性组合，如：
。视觉上可以被理解为：当固定某个标量，使另一个标量在实数范围内移动，则形成一条直线；当两个向量共线时，产生向量的终点在一条过原点的直线上。
2.4 张成的空间
定义：某两个向量的线性组合的向量之集合被称之为这两个向量张成的空间。
2.5 关于三维空间
两个三维向量：两个三维向量张成的空间是什么样的？一个过原点的平面。三个三维向量：当我们缩放第三个向量时，它将前两个向量张成的平面沿它的方向来回移动，从而扫过整个空间。
2.6 线性相关
定义：当第三个向量已经落在前两个向量张成的空间中（或者更低维度），而没有对张成空间做出任何贡献时，我们称它们线性相关。
三 矩阵与线性变换
3.1 线性变换的图像依据
对线性变换的直观理解。变换，变换的本质是运动，即一个向量移动到另一个向量的位置。线性，原点不动，并保持网格线平行且等距分布的变换。
线性变换的逻辑：一个向量是两个基向量的线性组合，当发生线性变换的时候，两个基向量发生同样的变换。此时，变换后的新向量就是变换后的基向量的线性组合，且两个标量与变换前相同。当原向量和变换方式已知，只需要知道变换方式，就存在以下式：
事实上，在平面中描述一个线性变化只需要一个22的矩阵（“2x2 Matrix”），第一列是变换后的i，第二列是变换后的*j。而计算方法很简单，思想如上。对于不管几维的矩阵都可以这样想，在这里思想是确定的。下面说一些特殊的矩阵
逆时针旋转90° 
剪切
本质上就是将(0,1)变换到(1,1) 然后x轴基不动。
四 矩阵乘法和线性变换复合
4.1 复合变换的几何意义
定义：先变换A矩阵，再变换B矩阵，称之为A和B矩阵的复合变换。
原理思考：我们可以定义一个C矩阵，用于描述先A再B的复合变换。本质上就是这句话；“新矩阵捕捉到了先A后B的相同的总体效应。”
注意： 矩阵乘法的本质意义是变换的顺序，因此当两个矩阵相乘的时候，需要从右往左读（因为g(f(x))是从左往右的，而右边的是先执行的内层函数）。
4.2 矩阵乘法
将右边的矩阵理解为变换完成的原向量即可，但请无比从右往左算，如下列式子：（“/”是自创符号，分割2x2矩阵的左右两列）
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由于这种连续变换的性质，先变换的矩阵在函数层级的内部，写在矩阵乘法的右边，优先计算。事实上，如有矩阵ABC，那么：
解释一下：矩阵乘法这个东西就是从右往左做的， 结合律就是没有影响…
4.3 三维空间中的线性变换
向量*矩阵，跟二维平面上的一样，如下：
%22%20aria-hidden%3D%22true%22%3E%0A%3Cg%20transform%3D%22translate(0%2C2156)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A1%22%20x%3D%220%22%20y%3D%22-1155%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(0%2C-2060.4867549668875)%20scale(1%2C0.5132450331125827)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A2%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A3%22%20x%3D%220%22%20y%3D%22-3167%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(834%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(-11%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-61%22%20x%3D%220%22%20y%3D%221356%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-64%22%20x%3D%223%22%20y%3D%22-44%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-67%22%20x%3D%2224%22%20y%3D%22-1450%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(1519%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-62%22%20x%3D%2273%22%20y%3D%221356%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-65%22%20x%3D%2255%22%20y%3D%22-44%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-68%22%20x%3D%220%22%20y%3D%22-1450%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(3095%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-63%22%20x%3D%2258%22%20y%3D%221356%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-66%22%20x%3D%220%22%20y%3D%22-44%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-69%22%20x%3D%22102%22%20y%3D%22-1450%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(4647%2C2156)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A4%22%20x%3D%220%22%20y%3D%22-1155%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(0%2C-2060.4867549668875)%20scale(1%2C0.5132450331125827)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A5%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A6%22%20x%3D%220%22%20y%3D%22-3167%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(5481%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(0%2C2153)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A1%22%20x%3D%220%22%20y%3D%22-1155%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(0%2C-2054.4966887417218)%20scale(1%2C0.5033112582781457)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A2%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A3%22%20x%3D%220%22%20y%3D%22-3161%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(834%2C0)%22%3E%0A%3Cg%20transform%3D%22translate(-11%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-78%22%20x%3D%220%22%20y%3D%221353%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-79%22%20x%3D%2237%22%20y%3D%22-47%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-7A%22%20x%3D%2252%22%20y%3D%22-1453%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(1563%2C2153)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A4%22%20x%3D%220%22%20y%3D%22-1155%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(0%2C-2054.4966887417218)%20scale(1%2C0.5033112582781457)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A5%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJSZ4-23A6%22%20x%3D%220%22%20y%3D%22-3161%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=%5Cbegin%7Bbmatrix%7Da%26b%26c%5C%5Cd%26e%26f%5C%5Cg%26h%26i%5C%5C%5Cend%7Bbmatrix%7D%0A%5Cbegin%7Bbmatrix%7Dx%5C%5Cy%5C%5Cz%5C%5C%5Cend%7Bbmatrix%7D%0A&id=jLJ5J)
矩阵*矩阵，本质上就是将之分解成向量*矩阵进行计算，不再提了。
五 行列式
5.1 概念
2D场景中
定义 行列式描述了在线性变换前后两个基向量围成的平行四边形面积的变化倍率。若两个基向量围成平行四边形的面积为A，在变换之后矩阵为
，平行四边形面积为
，线性变换的行列式写作
当行列式=0的时候，表明这个图形被压缩到一个更低维度的图形中——在2D场景中的线性相关的充要条件就是行列式等于0。
当行列式<0的时候，空间定向会发生翻转，在图像上与将坐标轴翻转等同。事实上从向量夹角的角度理解，i-hat和j-hat的夹角从正数到逐渐减小到0，再减小到负数区域（类比正弦函数）。
3D场景中
定义 线性变换前后三个基向量围成的平行六面体的体积的变化倍率。
当行列式=0的时候，整个空间被压缩为零体积模型（平面或直线）在这一条件下，矩阵的三个基向量线性相关。
当行列式<0的时候，判断方式如下：变换前使用右手定则，右手食指i-hat，中指j-hat，拇指k-hat；若变换后只能使用左手定则，即左手食指i-hat，中指j-hat，拇指k-hat，其行列式<0。**在几何上相当于把立方体的内部翻到外部**。
5.2 用矩阵计算行列式
2D场景中：
3D场景中：
2D的场景我能理解，但是这个逻辑能否推广到三维空间中？以及这些东西为什么存在这样的计算？
5.3 QuizTime
看以下式子：
，由于行列式实际上表示线性变换前后面积的变换倍率，因此变换倍率的乘积和线性变换后在计算面积变换率本质上没有差别。
六 逆矩阵、列空间与零空间
6.0 什么?
6.1 计算线性方程组
让我们来看一个相对简单的线性方程组：
对于这个方程组，存在两种可能性，
，此时空间并未挤压为0面积的区域，有且仅有一个向量在变换后会与v重合，我们可以通过逆向变换来找到这个向量——
事实上，
这是一个什么都不做的变换。那么我该怎么求得
呢？无论在2维平面还是在3维空间，只要矩阵的行列式不等于0，那么向量变换就存在唯一解，就能够用这种做法。
如果存在矩阵行列式等于0，见6.2
6.2 秩
定义：指变换后空间的维数。如果变换后的向量落在某个二维平面上，那么这个变换的秩为2；如果变换后的向量落在一个数轴上，那么这个变换的秩为1。
满秩：秩与列数相等。对于满秩变化而言，原点不变；但对于那些非满秩的变化而言，要么有体积被压缩成的平面，要么存在一个面被压缩到了一条直线。
6.3 列空间
列空间就是基向量在变化后张成的空间（所有可能性的集合），而更精确的秩的定义是列空间的维数。
6.4 零空间
6.5 非方阵【没懂】
有待补充
定义
输入输出空间不同的矩阵称为非方阵，比如3x2的矩阵虽然只有两个向量，但是其几何意义是存在于三维空间中的。
七 点积与对偶性
7.1 高中时期的向量点积
高中时期的向量点积有两种计算方式，如已知向量
，如已知两个向量 的坐标，则有：
那么为什么这两种计算方式是一样的？
7.2 对偶性
一句话：这个部分在几何直观上证明了1x2矩阵作为线性变换和一个2x1矩阵作为向量的同一性。
这个视频中还有很多值得思考的事情，建议再好好看看视频。
八 叉积的标准介绍
8.1 定义
引子：
围成的平行四边形的面积（这真的不是行列式吗）两个向量的顺序和最后的结果的正负性有关，当
在
右侧时，叉积的值为正数；当
在
左侧时，叉积的值为负数。
真正的叉积：通过两个三维向量生成一个新的三维向量。
定义：但事实上，叉积的结果是一个向量。这个向量的长度是变换后的行列式绝对值，方向垂直于这个平面，遵守右手定则（若叉积的结果是
，食指指向
，中指指向
，拇指指向
）
8.2 计算方式
设定一个2x2的矩阵M，将
作为第一列，将
作为第二列，然后将这个矩阵求行列式得出面积，最后根据两个向量的位置判断叉积的正负性。
正式的叉积计算方式如下：
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but why？？？下一集将解决i、j、k作为矩阵元所表达的含义和与几何直观。
八 通过线性变换深入理解叉积
8.3 叉积的几何结论
事实上，叉积获得的第三个向量
长度上等于前两个向量组成的平行四边形面积， 方向上垂直于这个平行四边形并满足右手定则。
8.4 几何特征的推导过程
对偶性快速回顾
当存在多维空间到数轴的线性变换时，它都与空间中的唯一向量对应。而应用线性变换和与这个向量点乘等价。数值上，这是因为这一类线性变换可以用一个只有一行的矩阵描述，而矩阵转置在计算上与另一个向量点乘相同。这个有矩阵转置的向量就是对偶向量。
根据
和
定义一个三维到一维的线性变换
来！
找到它的对偶向量
来！
证明这个对偶向量就是v和w的叉积
这个思路我确实没懂，所以还是打算放一个bv号在这里（p12），晚上回家再看看也无所谓。
九 基变换
基变换实际讲述了在坐标系之间对单个向量的描述进行相互转化，感觉本质上就是中译英、英译中的差别。用到的基础方法仍然是矩阵乘法，只不过要注意转化词典的问题。
9.1 英译中
有一对基
，若在此坐标系中有向量
，那么在2d空间中存在的相同的向量在平面直角坐标系中，其向量是
本质上，v向量是这两对基为标准的向量，M矩阵是以我们的语言描述的对应变换，因此要将v向量转换为我们的语言，将之与M词典做矩阵乘法即可。
9.2中译英
我们视角下（平面直角坐标系）的某个向量，如果想看看另一个视角下对这个向量的描述，则需要乘对应矩阵的逆。
十 特征值与特征向量
10.1 概念
10.2 应用：特征向量变换基
求解矩阵的多次方。
咕咕咕……
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