# 线性代数作业代写linear algebra代考|Dot product

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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

## 线性代数作业代写linear algebra代考|The negative of a vector

If $X=\left[\begin{array}{l}a_{1} \ b_{1} \ c_{1}\end{array}\right]$ and $Y=\left[\begin{array}{l}a_{2} \ b_{2} \ c_{2}\end{array}\right]$, then $X \cdot Y$, the dot product of $X$ and $Y$, is defined by $$X \cdot Y=a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2} .$$ 8.3. DOT PRODUCT 157 Figure 8.6: (a) Equality of vectors; (b) Addition and subtraction of (a) Equality of vectors; (b) Addition and subtraction of vectors. (i) $X \cdot(Y+Z)=X \cdot Y+X \cdot Z$; (ii) $X \cdot Y=Y \cdot X$; (iii) $(t X) \cdot Y=t(X \cdot Y)$; (iv) $X \cdot X=a^{2}+b^{2}+c^{2}$ if $X=\left[\begin{array}{l}a \ b \ c\end{array}\right]$ (v) $X \cdot Y=X^{t} Y$; (vi) $X \cdot X=0$ if and only if $X=0$. The length of $X$ is defined by $$|X|=\sqrt{a^{2}+b^{2}+c^{2}}=(X \cdot X)^{1 / 2}$$ distance between $P_{1}$ and $P_{2}$.

## 线性代数作业代写linear algebra代考|THREE–DIMENSIONAL GEOMETRY

Figure 8.7: Position vector as a linear combination of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. Vectors having unit length are called unit vectors. The vectors $$\mathbf{i}=\left[\begin{array}{l} 1 \ 0 \ 0 \end{array}\right], \quad \mathbf{j}=\left[\begin{array}{l} 0 \ 1 \ 0 \end{array}\right], \quad \mathbf{k}=\left[\begin{array}{l} 0 \ 0 \ 1 \end{array}\right]$$ are unit vectors. Every vector is a linear combination of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ : $$\left[\begin{array}{l} a \ b \ c \end{array}\right]=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$$ (See Figure 8.7.) It is easy to prove that $$|t X|=|t| \cdot|X|$$ if $t$ is a real number. Hence if $X$ is a non-zero vector, the vectors $$\pm \frac{1}{|X|} X$$ are unit vectors. A useful property of the length of a vector is $$|X \pm Y|^{2}=|X|^{2} \pm 2 X \cdot Y+|Y|^{2} .$$

## 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