把变换矩阵
A
A
A 看作两个二维列向量的组合
A
=
[
α
1
,
α
2
]
A=[\alpha_1,\alpha_2]
A=[α1,α2]，
α
1
,
α
2
\alpha_1,\alpha_2
α1,α2 称作此变换的基向量，
A
=
[
1
0.25
0
1
]
A=\begin{bmatrix} 1&0.25\\ 0&1 \end{bmatrix}
A=[100.251]的基向量为
α
1
=
[
1
0
]
\alpha_1=\begin{bmatrix} 1\\ 0\end{bmatrix}
α1=[10]和
α
2
=
[
0.25
1
]
\alpha_2=\begin{bmatrix} 0.25\\ 1\end{bmatrix}
α2=[0.251]，对于任意点
x
=
[
a
b
]
x=\begin{bmatrix} a\\ b\end{bmatrix}
x=[ab] 相乘表示两个基向量的线性组合
y
=
A
∗
x
=
[
α
1
,
α
2
]
[
a
b
]
=
a
α
1
+
b
α
2
=
a
[
1
0
]
+
b
[
0.25
1
]
=
[
a
+
0.25
b
b
]
y=A*x=[\alpha_1,\alpha_2]\begin{bmatrix} a\\ b\end{bmatrix}=a\alpha_1+b\alpha_2=a\begin{bmatrix} 1\\ 0\end{bmatrix}+b\begin{bmatrix} 0.25\\ 1\end{bmatrix}=\begin{bmatrix} a+0.25b\\ b\end{bmatrix}
y=A∗x=[α1,α2][ab]=aα1+bα2=a[10]+b[0.251]=[a+0.25bb]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
d
e
t
(
A
)
=
d
e
t
(
[
1
0.25
0
1
]
)
=
1
det(A)=det(\begin{bmatrix} 1&0.25\\ 0&1 \end{bmatrix})=1
det(A)=det([100.251])=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
=
[
0.866
−
0.5
0.5
0.866
]
A=\begin{bmatrix} 0.866&-0.5\\ 0.5&0.866 \end{bmatrix}
A=[0.8660.5−0.50.866]
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
