【数理知识】向量的坐标基表示法，Matlab 代码验证

序号	内容
1	【数理知识】向量的坐标基表示法，Matlab 代码验证
2	【数理知识】向量与基的内积，Matlab 代码验证

1. 向量的坐标基表示

假设空间中存在一个向量 $\vec{a}$，在不同的坐标系（或称坐标基）下，向量 $\vec{a}$ 由不同的坐标值表示。

当坐标基唯一确定后，对应的坐标值也唯一确定。同时向量也可以由坐标值和坐标基线性组合的形式来表示。

• 当向量为二维平面的向量时，可表示为
  $$\vec{a}=a_x{\vec{e}}_1+a_y{\vec{e}}_2=\left[\begin{array}{cc}{\vec{e}}_1& {\vec{e}}_2\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]$$

• 当向量为三维空间的向量时，可表示为
  $$\vec{a}=a_x{\vec{e}}_1+a_y{\vec{e}}_2+a_z{\vec{e}}_3=\left[\begin{array}{ccc}{\vec{e}}_1& {\vec{e}}_2& {\vec{e}}_3\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\\ a_z\end{array}\right]$$

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2. 二维平面向量举例

接下来基于二维平面的一个向量来举例，不过三维空间的情况具有同样的性质和结论。

假设存在一个上述的二维平面向量 $\vec{a}$，在标准坐标基 ${\vec{e}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right],{\vec{e}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$ 下的坐标值为 $\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$。那么此向量可以表示为

$$\begin{array}{rl}\vec{a}& =a_x{\vec{e}}_1+a_y{\vec{e}}_2=\left[\begin{array}{cc}{\vec{e}}_1& {\vec{e}}_2\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]\\ & =3\left[\begin{array}{c}1\\ 0\end{array}\right]+4\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]\end{array}$$

现在，我们更改坐标基为 ${\vec{e}}_{1^{\prime }}=\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right],{\vec{e}}_{2^{\prime }}=\left[\begin{array}{c}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]$，此新基下的坐标值为 $\left[\begin{array}{c}{a}_{x^{\prime }}\\ a_{y^{\prime }}\end{array}\right]=\left[\begin{array}{c}\frac{7}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]$。那么此向量可以表示为

$$\begin{array}{rl}\vec{a}& =\frac{7}{\sqrt{2}}\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]+\frac{1}{\sqrt{2}}\left[\begin{array}{c}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{c}\frac{7}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]\end{array}$$

上述例子我们可以看到，无论是在哪个坐标基下，永远存在如下等式

$$\begin{array}{rl}\vec{a}& =a_x{\vec{e}}_1+a_y{\vec{e}}_2=\left[\begin{array}{cc}{\vec{e}}_1& {\vec{e}}_2\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]\\ & =a_{x^{\prime }}{\vec{e}}_{1^{\prime }}+a_{y^{\prime }}{\vec{e}}_{2^{\prime }}=\left[\begin{array}{cc}{\vec{e}}_{1^{\prime }}& {\vec{e}}_{2^{\prime }}\end{array}\right]\left[\begin{array}{c}{a}_{x^{\prime }}\\ a_{y^{\prime }}\end{array}\right]\end{array}$$

针对三维空间中的向量，同样具有类似的结论。

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至于新基下的坐标值是如何得到的，我们可以通过以下步骤实现

$$\begin{array}{rl}\left[\begin{array}{cc}{\vec{e}}_1& {\vec{e}}_2\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]& =\left[\begin{array}{cc}{\vec{e}}_{1^{\prime }}& {\vec{e}}_{2^{\prime }}\end{array}\right]\left[\begin{array}{c}{a}_{x^{\prime }}\\ a_{y^{\prime }}\end{array}\right]\\ {\left[\begin{array}{cc}{\vec{e}}_{1^{\prime }}& {\vec{e}}_{2^{\prime }}\end{array}\right]}^{-1}\left[\begin{array}{cc}{\vec{e}}_1& {\vec{e}}_2\end{array}\right]\left[\begin{array}{c}{a}_x\\ a_y\end{array}\right]& =\left[\begin{array}{c}{a}_{x^{\prime }}\\ a_{y^{\prime }}\end{array}\right]\\ {\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]}^{-1}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]& =\left[\begin{array}{c}\frac{7}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right]\end{array}$$

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3. Matlab 代码验证

a_x = 3;
a_y = 4;

e_1 = [ 1
        0];
e_2 = [ 0
        1];

a_x_prime = 7/sqrt(2);
a_y_prime = 1/sqrt(2);

e_1_prime = [ sqrt(2)/2
              sqrt(2)/2];
e_2_prime = [-sqrt(2)/2
              sqrt(2)/2];

>> pinv([e_1_prime  e_2_prime]) * [e_1  e_2] * [a_x; a_y]
ans =
    4.9497
    0.7071

>> a_x_prime
ans =
    4.9497

>> a_y_prime
ans =
    0.7071

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Ref

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