matlab分析矩阵与线性变换

• 把变换矩阵 $A$ 看作两个二维列向量的组合 $A=[{\alpha }_1,{\alpha }_2]$，${\alpha }_1,{\alpha }_2$ 称作此变换的基向量，$A=\left[\begin{array}{cc}1& 0.25\\ 0& 1\end{array}\right]$的基向量为${\alpha }_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$和${\alpha }_2=\left[\begin{array}{c}0.25\\ 1\end{array}\right]$，对于任意点 $x=\left[\begin{array}{c}a\\ b\end{array}\right]$ 相乘表示两个基向量的线性组合
  $$y=A\ast x=[{\alpha }_1,{\alpha }_2]\left[\begin{array}{c}a\\ b\end{array}\right]=a{\alpha }_1+b{\alpha }_2=a\left[\begin{array}{c}1\\ 0\end{array}\right]+b\left[\begin{array}{c}0.25\\ 1\end{array}\right]=\left[\begin{array}{c}a+0.25b\\ b\end{array}\right]$$
• 行列式的几何意义是两个基向量所构成平行四边形的面积，新图形变换后的面积的倍数等于行列式的绝对值，负数代表图形翻转
  $$det(A)=det(\left[\begin{array}{cc}1& 0.25\\ 0& 1\end{array}\right])=1$$

x1 = [0;0];
x2 = [1;0];
x3 = [1;1];
x4 = [0;1];
x = [x1,x2,x3,x4,x1];
A = [1 0.25;0 1];
y = A * x;
plot(x(1,:),x(2,:), 'b')
hold on
plot(y(1,:),y(2,:),'r')
axis([0 1.5 0 1.5])
grid on

• 旋转的效果 $A=\left[\begin{array}{cc}0.866& -0.5\\ 0.5& 0.866\end{array}\right]$

x1 = [0;0];
x2 = [1;0];
x3 = [1;1];
x4 = [0;1];
x = [x1,x2,x3,x4,x1];
A = [0.866 -0.5;0.5 0.866];
y = A * x;
plot(x(1,:),x(2,:), 'b')
hold on
plot(y(1,:),y(2,:),'r')
axis([-1 1.5 0 1.5])
grid on