# 线性代数作业代写linear algebra代考|Show that the planes

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

$$x+y-2 z=1 \quad \text { and } \quad x+3 y-z=4$$ intersect in a line and find the distance from the point $C=(1,0,1)$ to this line. Solution. Solving the two equations simultaneously gives $$x=-\frac{1}{2}+\frac{5}{2} z, \quad y=\frac{3}{2}-\frac{1}{2} z,$$ where $z$ is arbitrary. Hence $$x \mathbf{i}+y \mathbf{j}+z \mathbf{k}=-\frac{1}{2} \mathbf{i}-\frac{3}{2} \mathbf{j}+z\left(\frac{5}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}\right),$$ which is the equation of a line $\mathcal{L}$ through $A=\left(-\frac{1}{2},-\frac{3}{2}, 0\right)$ and having direction vector $\frac{5}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}$.

We can now proceed in one of three ways to find the closest point on $\mathcal{L}$ to $A$. One way is to use equation $8.17$ with $B$ defined by $$\overrightarrow{A B}=\frac{5}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}$$ Another method minimizes the distance $C P$, where $P$ ranges over $\mathcal{L}$. A third way is to find an equation for the plane through $C$, having $\frac{5}{2} \mathbf{i}-\frac{1}{2} \mathbf{j}+\mathbf{k}$ as a normal. Such a plane has equation $$5 x-y+2 z=d,$$ where $d$ is found by substituting the coordinates of $C$ in the last equation. $$d=5 \times 1-0+2 \times 1=7 .$$ We now find the point $P$ where the plane intersects the line $\mathcal{L}$. Then $\overrightarrow{C P}$ will be perpendicular to $\mathcal{L}$ and $C P$ will be the required shortest distance from $C$ to $\mathcal{L}$. We find using equations $8.27$ that $$5\left(-\frac{1}{2}+\frac{5}{2} z\right)-\left(\frac{3}{2}-\frac{1}{2} z\right)+2 z=7$$

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions