课程： 7.2 一套正弦矢量

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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。

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。

Theorem: If $\begin{aligned} S \end{aligned}$ is a set of orthogonal vectors in $\begin{aligned} R^{n} \end{aligned}$ , then $\begin{aligned} S \end{aligned}$ 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。

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。

Definition: An orthogonal basis for a subspace $\begin{aligned} W of R^{n} \end{aligned}$ is a basis for $\begin{aligned} W \end{aligned}$ 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。

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。

Theorem: If we have an orthogonal basis of the form $\begin{aligned} {{\vec{u}}_{1}, {\vec{u}}_{2}, \dots, {\vec{u}}_{n}} \end{aligned}$ 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

$\begin{aligned} \vec{v} = (\frac{\vec{v} \cdot \vec{u_{1}}}{\vec{u_{1}} \cdot \vec{u_{1}}}) \cdot \vec{u_{1}} + \dots + (\frac{\vec{v} \cdot \vec{u_{n}}}{\vec{u_{n}} \cdot \vec{u_{n}}}) \cdot \vec{u_{n}} \end{aligned}$
::

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

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

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

$\begin{aligned} [\begin{matrix} 3 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} - 1 \\ 2 \\ 1 \end{matrix}], [\begin{matrix} \frac{- 1}{2} \\ - 2 \\ \frac{7}{2} \end{matrix}] \end{aligned}$

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。

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

Let's label these vectors $\begin{aligned} \vec{u_{1}}, \vec{u_{2}}, \vec{u_{3}} \end{aligned}$
::让我们标记这些矢量 u1,u2,u3

Now, let's say we want to find the vector $\begin{aligned} \vec{v} = [\begin{matrix} 6 \\ 1 \\ - 8 \end{matrix}] \end{aligned}$ 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:

$\begin{aligned} \vec{v} \cdot \vec{u_{1}} = 11 \\ \vec{v} \cdot \vec{u_{2}} = - 12 \\ \vec{v} \cdot \vec{u_{3}} = - 33 \\ \vec{u_{1}} \cdot \vec{u_{1}} = 11 \\ \vec{u_{2}} \cdot \vec{u_{2}} = 6 \\ \vec{u_{3}} \cdot \vec{u_{3}} = \frac{33}{2} \\ \vec{v} = \vec{u_{1}} - 2 \vec{u_{2}} - 2 \vec{u_{3}} \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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。

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

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

$\begin{aligned} \hat{y} = \frac{\vec{y} \cdot \vec{u}}{\vec{u} \cdot \vec{u}} \cdot \vec{u} \end{aligned}$
::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

as the orthogonal projection of $\begin{aligned} \vec{y} onto \vec{u} . \end{aligned}$
::作为 y 的正向投影到 u 的 y 垂直投影。

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:

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。

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。

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

Let $\begin{aligned} W \end{aligned}$ be a subspace of $\begin{aligned} R^{n} \end{aligned}$ and $\begin{aligned} \vec{y} \in R^{n} \end{aligned}$ and $\begin{aligned} \hat{y} = {proj}_{W} (\vec{y}) \end{aligned}$ then $\begin{aligned} \hat{y} \end{aligned}$ is the point in W closest to $\begin{aligned} \vec{y} \end{aligned}$ for every single $\begin{aligned} \vec{v} \in W \end{aligned}$ not equal to $\begin{aligned} \hat{y} \end{aligned}$ .
::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。

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。

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。

章节大纲

常规

整形矢量集