Introdunction to Linear Algebra-第1章-Introduction to Vectors

[TOC]
《Introdunction to Linear Algebra》的第 1 章：Introduction to Vectors.

1.1 Vector addition $\boldsymbol{v} + \boldsymbol{w}$ and linear combinations $c\boldsymbol{v} + d\boldsymbol{w}$.
1.2 The dot product $\boldsymbol{v} · \boldsymbol{w}$ of two vectors and the length $||\boldsymbol{v}|| =\sqrt{\boldsymbol{v} \boldsymbol{v}}$
1.3 Matrices $A$, linear equations $A\boldsymbol{x} = \boldsymbol{b}$, solutions $\boldsymbol{x} = A^{- 1} \boldsymbol{b}$.

1.1 Vectors and Linear Combinations

• vector addition
• scalar multiplication

Represent vector $\boldsymbol{v}$

1. Two numbers
2. Arrow from $(0, 0)$
3. Point in the plane

1.2 Lengths and Dot Products

三角不等式

A unit vector $\boldsymbol{w}$ is a vector whose length equals one. Then $\boldsymbol{w} · \boldsymbol{w} = 1$.
Unit vector $\boldsymbol{w} = \boldsymbol{v} / || \boldsymbol{v} ||$ is a unit vector in the same direction as $\boldsymbol{v}$.
$$
\text { Unit vectors } \quad \boldsymbol{i}=\left[\begin{array}{l}
1 \\
0
\end{array}\right] \quad \text { and } \quad \boldsymbol{j}=\left[\begin{array}{l}
0 \\
1
\end{array}\right] \quad \text { and } \quad \boldsymbol{u}=\left[\begin{array}{c}
\cos \theta \\
\sin \theta
\end{array}\right]
$$

Unit vectors $\boldsymbol{w}$ and $U$ at angle $\theta$ have $\boldsymbol{w} · U = cos \theta$. Certainly $||\boldsymbol{w} · U|| \leq 1$.

COSINE FORMULA
If $\boldsymbol{v}$ and $\boldsymbol{w}$ are nonzero vectors then $\frac{\boldsymbol{v} \cdot \boldsymbol{w}}{||\boldsymbol{v}|| ||\boldsymbol{w}||}=cos \theta$.

SCHWARZ INEQUALITY
$$
||\boldsymbol{v} \cdot \boldsymbol{w}|| \leq ||\boldsymbol{v}|| ||\boldsymbol{w}||
$$

TRIANGLE INEQUALITY
$$
|| \boldsymbol{v} + \boldsymbol{w} || \leq || \boldsymbol{v} || + || \boldsymbol{w} ||
$$

$$
\begin{array}{ccc}
\text { Geometric } & \leq \text { Arithmetic } \\
\text { mean } & \text { mean }
\end{array} \quad a b \leq \frac{a^{2}+b^{2}}{2} \quad \text { becomes } \quad \sqrt{x y} \leq \frac{x+y}{2}
$$

1.3 Matrices

2种理解矩阵的方式

1. 列的 linear combination
   the column picture of $A\boldsymbol{x}=\boldsymbol{b}$: a combination of $n$ columns of $A$ produces the vector $\boldsymbol{b}$.
2. 行乘以列的 inner product
   the raw picture of $A\boldsymbol{x}=\boldsymbol{b}$: $m$​ equations from $m$​ rows give $m$​ planes meeting at $\boldsymbol{x}$​.

$$
\begin{aligned}
&A \boldsymbol{x} \text { is also } \\
&\text { dot products } \\
&\text { with rows }
\end{aligned} A \boldsymbol{x}=\left[\begin{array}{rrr}
1 & 0 & 0 \\
-1 & 1 & 0 \\
0 & -1 & 1
\end{array}\right]\left[\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right]=\left[\begin{array}{l}
(1,0,0) \cdot\left(x_{1}, x_{2}, x_{3}\right) \\
(-1,1,0) \cdot\left(x_{1}, x_{2}, x_{3}\right) \\
(0,-1,1) \cdot\left(x_{1}, x_{2}, x_{3}\right)
\end{array}\right]
$$

$(\textbf{row} : 1) \cdot x = b_1,…,(\textbf{row} : m) \cdot x = b_m$​
前者是理解线性代数本质的关键, 后者用于计算.

The Inverse Matrix

矩阵与其逆可以与微积分里的积分与微分做类比:

$$
\boldsymbol{A} \boldsymbol{x}=\boldsymbol{b} \text { and } \boldsymbol{x}=\boldsymbol{A}^{-1} \boldsymbol{b} \quad \frac{d x}{d t}=b \text { and } x(t)=\int_{0}^{t} b d t
$$

Independence and Dependence

线性相关与线性无关.

线性相关的例子,$\boldsymbol{w}, \boldsymbol{v}, \boldsymbol{w}$ 张成一个平面 $\mathbf{R}^2$,而不是三维的$\mathbf{R}^3$.

Independent columns: $A\boldsymbol{x} = 0$ has one solution. $A$ is an invertible matrix.
Dependent columns: $C\boldsymbol{x} = 0$ has many solutions. $C$ is a singular matrix.