参数矢量形式的解决方案

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In the past when we have solved matrix-vector equations or systems of linear equations we just declare a variable free or declare infinitely many solutions whenever we have infinite solutions.
::过去,当我们解决了矩阵-矢量方程式或线性方程式系统时,我们只是宣布一个变量自由,或者只要我们有无限的解决办法,就会宣布无限多的解决办法。

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However, we are now going to introduce a new way to articulate the solution. We are going to use something called parametric vector form.
::然而,我们现在将引入一种新的方法来解释解决方案。我们将使用一种叫做参数矢量表的方法。

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Consider the system of equations:
::考虑方程体系:

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$\begin{align*}3x_{1} - 2x_{2} + 4x_{3} = 7\\ -x_{1} + 5x_{2} - 6x_{3} = 2\\ 0x_{1} + 0x_{2} + 0x_{3} = 0\\ \end{align*}$
::3x1-2x2+4x3=7-x1+5x2-6x3=20x1+0x2+0x3=0

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Writing that as a matrix-vector equation you get
::以矩阵矢量方程式的形式写

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$\begin{align*}\begin{bmatrix}3&-2&4\\-1&5&-6\\0&0&0\end{bmatrix}\cdot\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix} = \begin{bmatrix}7\\2\\0\end{bmatrix}\end{align*}$
::[3-24-15-6000]/[x1x2x3]=[720]

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This turns into the augmented matrix 
::转而成为扩充的矩阵表

$\begin{align*}\begin{bmatrix}3 & -2 & 4&\mid& 7\\-1&5&-6&\mid& 2\\0&0&0&\mid&0\end{bmatrix}\end{align*}$

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Applying row reduction comes out to
::应用换行

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$$\begin{align*}\begin{bmatrix} 3 & -2 & 4 & \mid & 7 \\ -1 & 5 & -6 & \mid & 2 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \begin{bmatrix} -1 & 5 & -6 & \mid & 2 \\ 3 & -2 & 4 & \mid & 7 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \begin{bmatrix} 1 & -5 & 6 & \mid & -2 \\ 3 & -2 & 4 & \mid & 7 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \begin{bmatrix} 1 & -5 & 6 & \mid & -2 \\ 0 & 13 & -14 & \mid & 13 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \begin{bmatrix} 1 & 0 & \frac{8}{13} & \mid & 3 \\ 0 & 13 & -14 & \mid & 13 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \begin{bmatrix} 1 & 0 & \frac{8}{13} & \mid & 3 \\ 0 & 1 & -\frac{14}{13} & \mid & 1 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\end{align*}$$ This translates back to the equation
::[1 - 56 - 23 - 24 - 24 _ 700_ 01][ 1 - 56 - 23 - 24 _ 7000_ 0][ 1 - 56 - 2013 - 14 @ 13000 _ 0][10813_ 3013 - 14_ 13_ 13_ 13_ 13_ 13_ 13_ 13_ 2000_ 0][10813_ 301 - 1413_ 1000_0]]

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   $\begin{align*}x_{1} - \frac{8}{13}x_{3} = 3\\ x_{2} - \frac{14}{13}x_{3} = 1\\ x_{3} = x_{3}\end{align*}$
::x1 - 813x3=3x2 - 1413x3=1x3=x3

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Simplifying to
::简化到

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  $\begin{align*}x_{1} = 3 + \frac{8}{13}x_{3}\\ x_{2} = 1 + \frac{14}{13}x_{3}\\ x_{3} = x_{3}\end{align*}$
::x1=3+813x3x2=1+1413x3x3=x3

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Now recalling that our solution vector  $\begin{align*}\vec{x} = \begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}\end{align*}$  we can substitute back in to get  $\begin{align*}\vec{x} = \begin{bmatrix}3+\frac{8}{13}x_{3}\\1+\frac{14}{13}x_{3}\\x_{3}\end{bmatrix}\\ \vec{x} = \begin{bmatrix}3+\frac{8}{13}x_{3}\\1+\frac{14}{13}x_{3}\\0+x_{3}\end{bmatrix}\\ \vec{x} = \begin{bmatrix}3\\1\\0\end{bmatrix}+x_{3}\begin{bmatrix}\frac{8}{13}\\\frac{14}{13}\\1\end{bmatrix}\\ \vec{x} = \begin{bmatrix}3\\1\\0\end{bmatrix}+\frac{x_{3}}{13}\begin{bmatrix}8\\14\\13\end{bmatrix}\end{align*}$
::现在回顾我们的溶性矢量 x[x1x2x3],我们可以重新替换,以获得 x[3+813x31+1413x3x3]xxx{3][3+813x31+1413x30+x3]x{[310]+x3[81314131]x}x}}[310]+x3[310]+314131]x}}[310]+x313[81413]x413]

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Geometrically, we see the solution set is a  line: 
::从几何上看,我们看到解决方案是一条线:

 

 

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Now let's look at the reduced row echelon form of the matrix that this system yields. We ended up getting
::现在让我们看看这个系统生成的矩阵的 减排梯层形式。我们最后得到的是

  $\begin{align*}\begin{bmatrix} 1 & 0 & \frac{8}{13} & \mid & 3 \\ 0 & 1 & -\frac{14}{13} & \mid & 1 \\ 0 & 0 & 0 & \mid & 0 \end{bmatrix}\\ \end{align*}$

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and there is no further way to reduced this. Basically, you cannot put this into the equivalent of the identity matrix to the left of the augment and there are only two pivot columns. A pivot entry is the entry corresponding to the leading one in the reduced row echelon form of the matrix and a pivot column is a column that contains a pivot. When you have three pivots, you get a point as it equates to a solution for all  $\begin{align*}x_{1},x_{2},x_{3},\cdots x_{n}\end{align*}$   which equates to a vector as a single point in space.
::并且没有进一步的方法去缩小这一点。 基本上, 您不能将此设置在与扩展左侧等同的身份矩阵中, 并且只有两条支流列。 一条支流条目是矩阵的减行梯层形式的前列条目, 一条支流列是包含一个支流的列。 当您有三个支流时, 您会得到一个点, 因为它相当于所有 x1, x2, x3, xxxxn 的解决方案, 它相当于一个矢量作为空间的单一点 。

 

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Essentially what you are doing when you are searching for a solution set is trying to get ones down the main diagonal and get a vector solution. However, it is not always the case that we end up getting those ones down the diagonal. 
::基本上,当你正在寻找一个解决方案集时,你正在做的就是试图把那些放在主对角,然后找到一个矢量解决方案。然而,我们并非总能最终把那些放到对角。

 

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Now, recall what we talked about with linear dependence relations among columns. If the columns form a linear dependence relation we can get the matrix into an echelon form where there will be some number of ones down the main diagonal that is not equal to the dimension of the matrix. Which ever columns are not pivot columns then have their corresponding variable as a free variable.
::现在,请记住我们谈论的列间线性依赖关系。如果列构成线性依赖关系,我们可以将矩阵变成一个梯层,在主对角下将有一些与矩阵维度不相等的矩阵。那些纵列永远不是支流列,然后将相应的变量作为自由变量存在。

 

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Looking at a case with a 3x3 matrix we have that no free variables gives a point, 1 gives a line, 2 give a plane, because 2 vectors span a plane and in a general  $\begin{align*}n\end{align*}$   case,  $\begin{align*}n\end{align*}$   free variables give whatever the  $\begin{align*}n\end{align*}$   dimensional analogue is.
::以 3x3 矩阵来查看一个案例, 我们发现没有自由变量给出一个点, 1 给出一条线, 2给出一平面, 因为 2 个矢量横跨一平面, 在一般的 n 情况下, n 自由变量给出任何 n 维类比 。

 

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Now, let's try to talk about this generally as an  $\begin{align*}m\times n\end{align*}$   case matrix. So if you have a matrix with more rows than columns, what you should do to get a better picture of what is going on is to happen you should add columns of 0's, which preserve the structure which make it an $\begin{align*}n\times n\end{align*}$  matrix and an  $\begin{align*}n\times n\end{align*}$   system of equations, so you can see how many solutions there are going to be and what the solution set will actually look like. 
::现在,让我们试着把这个问题一般地说成一个 mxn 案例矩阵。因此,如果你有一个比列多行的矩阵,那么,你应该做些什么才能更清楚地了解正在发生的事情,你应该增加0's的列,这保留了使它成为 nxn 矩阵和 nxn 方程体系的结构,这样你就可以看到有多少解决方案将会存在,以及解决方案集实际上会是什么样子。

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Now be careful with this case as I'll illustrate with this example:
::现在请小心处理这个案件,因为我要用这个例子来说明:

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$\begin{align*}2x+3y=5\\ 3x+5y=8\\ 4x+6y=10\\ \begin{bmatrix}2 & 3&\mid&5\\3&5&\mid&8\\4&6&\mid&10\end{bmatrix}\\ \begin{bmatrix}2 & 3&0&\mid&5\\3&5&0&\mid&8\\4&6&0&\mid&10\end{bmatrix}\end{align*}$  .
::2x+3y=53x+5y=84x+6y=10[23\535\846\10][230\5350\8460\10]。

 

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Solving this would give you $\begin{align*}x=1\text{ and }y=1.\end{align*}$  It would not give you a third free variable, because reducing it would give you a row of 0's. Basically, be aware of redundant equations. 
::解决这个问题会给您 x=1 和 Y=1 。 它不会给您第三个自由变量, 因为减少它会给您一列 0 。 基本上, 了解多余的方程式 。

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There is also the opposite case for  $\begin{align*}m\times n\end{align*}$  matrices where you can add rows of zeros. When you add rows of zeros you basically get this to be an equation of more variables, but that has a bunch of extra variables as 0.
::mxn 矩阵也有相反的情况,您可以在此添加零行。当您添加零行时,您基本上会把它变成由更多变量组成的方程式,但有一组额外的变量作为 0 。

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Essentially, when a variable is 0 we can just ignore it. Think about it this way. The equations you deal with in 2 dimensions or a two dimensional vector is really just a vector existing in 3d space, but on the two dimensional x-y plane. The same goes for any dimensions there are just a bunch of zeros we don't use. 
::基本上, 当变量为 0 时, 我们就可以忽略它。 这样考虑它。 您处理的 2 维或 2 维矢量的方程式, 实际上是 3D 空格中存在的矢量, 但是在 2 维 x - y 平面上存在 。 对于任何维值来说, 都是一样的。 我们没有使用这些公式 。

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In the first example we did, we essentially had two equations as the bottom was a row of zeros. You, however, had to add the row of zeros in order for there to really be a third variable and the line to exist. The second vector with the  $\begin{align*}x_{3}\end{align*}$  coefficient was in three dimensions and if this system stated with two rows it would have been possible to solve but not very visually comprehensible.
::在第一个例子中,我们基本上有两个方程,因为底部是零的行。然而,你必须加上零的行,这样才能真正成为第三个变量和行。x3系数的第二个矢量有三个维度,如果这个系统用两行来表示,它本来是可以解决的,但不能非常直观地理解。

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All in all, the goal with these types of problems is to give yourself the best visual as possible. Usually you will be dealing with 3 dimensional equations for rigorous applications of physics and engineering (though sometimes you can deal with abstract high dimensional vectors).
::总而言之,这些问题总而言之,目标是给自己尽可能最好的视觉效果。 通常你将处理物理学和工程学严格应用的三维方程(尽管有时你可以处理抽象的高维矢量 ) 。

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