7.2 一套正弦矢量

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  整形矢量集

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  In this lesson we are going to be learning about sets of orthogonal vectors and their properties.
  ::在这个教训中,我们将学习 几组正向矢量及其属性。

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  Definition: A set of vectors is a set of orthogonal vectors if and only if each vector is orthogonal with one another.
  ::定义:一组矢量是指一套正对角矢量,条件是且仅在每个矢量相互对齐的情况下。

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  Theorem: If  $\begin{array}{r}S\end{array}$  is a set of orthogonal vectors in  $\begin{array}{r}{\mathbb{R}}^n\end{array}$ , then  $\begin{array}{r}S\end{array}$  is linearly independent and thus can be treated as a basis for the subspace spanned by those orthogonal vectors.
  ::论理: 如果 S 是 Rn 中一组正向矢量, 那么 S 是线性独立的, 因此可以作为这些正向矢量所跨越的子空间的基础 。

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  As an exercise try and prove this theorem on your own.
  ::作为锻炼 尝试证明这个理论 你自己的。

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  Definition: An orthogonal basis for a subspace  $\begin{array}{r}W\text{ of }{\mathbb{R}}^n\end{array}$  is a basis for  $\begin{array}{r}W\end{array}$  which also happens to be an orthogonal set.
  ::定义:Rn 子空间W的正弦基是W的基础,它也恰好是一个正弦基。

   

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  What's especially nice about orthogonal basis is we can exploit their properties to find an easy way to calculate the coefficients of vectors when taking a linear combination of them.
  ::最有趣的是,我们可以利用它们的特性 来找到一个简单的方法 来计算矢量的系数, 用它们的线性组合来计算。

   

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  Theorem: If we have an orthogonal basis of the form  $\begin{array}{r}\{{\stackrel{\to }{u}}_1,{\stackrel{\to }{u}}_2,\cdots ,{\stackrel{\to }{u}}_n\}\end{array}$  then given an element that is in the subspace we can write 
  ::定理: 如果我们对表 {u1, u2, , , , } 的正对线基, 那么给定一个元素, 元素在子空间中, 我们可以写入

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  $\begin{array}{r}\stackrel{\to }{v}=(\frac{\stackrel{\to }{v}\cdot \stackrel{\to }{u_1}}{\stackrel{\to }{u_1}\cdot \stackrel{\to }{u_1}})\cdot \stackrel{\to }{u_1}+\cdots +(\frac{\stackrel{\to }{v}\cdot \stackrel{\to }{u_n}}{\stackrel{\to }{u_n}\cdot \stackrel{\to }{u_n}})\cdot \stackrel{\to }{u_n}\end{array}$
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  Al though this theorem is a bit harder to prove, I have faith that you can prove it on your own so try and prove it as an exercise.
  ::虽然这个定理比较难证明 但我相信你可以自己证明 所以试着把它证明为一种练习

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  Let's now look at some examples of orthogonal basis :
  ::让我们看看一些垂直基础的例子:

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  Let S be the set spanned by the vectors
  ::让 S 成为矢量设定的宽度

    $\begin{array}{r}\left[\begin{array}{c}3\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\right],\left[\begin{array}{c}\frac{-1}{2}\\ -2\\ \frac{7}{2}\end{array}\right]\end{array}$

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  We verify that this is an orthogonal basis by seeing that 
  ::我们通过看到这一点来核实这是否是一个正正向基础。

  $\begin{array}{r}(3,1,1)\cdot (-1,2,1)=0\\ (3,1,1)\cdot (-1/2,2,7/2)=0\\ (-1,2,1)\cdot (-1/2,2,7/2)=0\end{array}$

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  Let's  label these vectors  $\begin{array}{r}\stackrel{\to }{u_1},\stackrel{\to }{u_2},\stackrel{\to }{u_3}\end{array}$
  ::让我们标记这些矢量 u1,u2,u3

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  Now, let's say we want to find the vector  $\begin{array}{r}\stackrel{\to }{v}=\left[\begin{array}{c}6\\ 1\\ -8\end{array}\right]\end{array}$  as a linear combination of the top three vectors. We  continue as follows:
  ::假设我们想找到矢量 v[61-8] 作为前三个矢量的线性组合。 我们继续如下:

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  $\begin{array}{r}\stackrel{\to }{v}\cdot \stackrel{\to }{u_1}=11\\ \stackrel{\to }{v}\cdot \stackrel{\to }{u_2}=-12\\ \stackrel{\to }{v}\cdot \stackrel{\to }{u_3}=-33\\ \stackrel{\to }{u_1}\cdot \stackrel{\to }{u_1}=11\\ \stackrel{\to }{u_2}\cdot \stackrel{\to }{u_2}=6\\ \stackrel{\to }{u_3}\cdot \stackrel{\to }{u_3}=\frac{33}{2}\\ \stackrel{\to }{v}=\stackrel{\to }{u_1}-2\stackrel{\to }{u_2}-2\stackrel{\to }{u_3}\end{array}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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  One can easily verify that the final result is true just by inspection.
  ::人们可以很容易地通过检查来核实最终结果是否属实。

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  Now, we'll introduce the notion of orthogonal projections.
  ::现在,我们来介绍 垂直预测的概念。

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  Here, we define the vector 
  ::在此, 我们定义矢量

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  $\begin{array}{r}\hat{y}=\frac{\stackrel{\to }{y}\cdot \stackrel{\to }{u}}{\stackrel{\to }{u}\cdot \stackrel{\to }{u}}\cdot \stackrel{\to }{u}\end{array}$
  ::你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你,你

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  as the orthogonal projection of  $\begin{array}{r}\stackrel{\to }{y}\text{ onto }\stackrel{\to }{u}.\end{array}$
  ::作为 y 的正向投影到 u 的 y 垂直投影 。

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  Now, try and make your own graphical interpretation of this:
  ::现在,试着用你自己的图形来解释这个:

   

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  This is essentially the projection of a vector onto another as we discussed in lesson 1.6. Go back to review more of  the properties of projections.
  ::这基本上是我们在第1.6项中所讨论的向量投射到另一个向量。 回去审查更多的预测属性。

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  This is also what the coefficients look like in the equation for a vector in the span of vectors in an orthogonal basis.
  ::这也是矢量方程中的矢量在矢量间距的正方程中的系数。

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  Here's a theorem to consider:
  ::这里有一个理论要考虑:

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  Let  $\begin{array}{r}W\end{array}$  be a subspace of  $\begin{array}{r}{\mathbb{R}}^n\end{array}$  and  $\begin{array}{r}\stackrel{\to }{y}\in {\mathbb{R}}^n\end{array}$  and  $\begin{array}{r}\hat{y}={\text{proj}}_W(\stackrel{\to }{y})\end{array}$  then  $\begin{array}{r}\hat{y}\end{array}$  is the point in W closest to  $\begin{array}{r}\stackrel{\to }{y}\end{array}$   for every single  $\begin{array}{r}\stackrel{\to }{v}\in W\end{array}$ not equal to  $\begin{array}{r}\hat{y}\end{array}$  .
  ::让 W 成为 Rn 和 y 和 y 和 projW 的子空间, 那么y 将是每个 v 和 y 和 y 最接近 y 的点 。

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  Again, try and think of a new visual interpretation of this.  
  ::再一次,试着想出一个新的视觉解释。

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  Now look at these videos and these links to help solidify your understanding of orthogonal projections and orthogonal sets of vectors.
  ::现在看看这些视频和这些链接, 帮助巩固您对正向投影和正向矢量组合的理解。