Chapter 1 (Linear Equations in Linear Algebra): Vector equations (向量方程)

本文为《Linear algebra and its applications》的读书笔记

Vectors in ${\mathbb{R}}^2$

• A matrix with only one column is called a column vector (列向量), or simply a vector (向量).
  $$\mathbf{w}=\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]$$where $w_1$ and $w_2$ are any real numbers.
• The set of all vectors with two entries is denoted by ${\mathbb{R}}^2$(read “r-two”). The $\mathbb{R}$ stands for the real numbers that appear as entries in the vectors, and the exponent 2 indicates that each vector contains two entries. Vectors in ${\mathbb{R}}^2$ are ordered pairs of real numbers. (实数的有序对)

Sometimes, for convenience (and also to save space), this text may write a column vector in the form $(w_1,w_2)$. In this case, the parentheses and the comma distinguish the vector $(w_1,w_2)$ from the $1\times 2$ row matrix $[w_1{w}_2]$, written with brackets and no comma.

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scalar multiple (标量乘法 / 数乘)：

Geometric Descriptions of ${\mathbb{R}}^2$

• Consider a rectangular coordinate system (直角坐标系) in the plane. We can identify a geometric point $(a,b)$ with the column vector $\left[\begin{array}{c}a\\ b\end{array}\right]$.
  
• The sum of two vectors has a useful geometric representation. The following rule can be verified by analytic geometry. (解析几何)
  
  

For simplicity of notation, a vector such as $\mathbf{u}+(-1)\mathbf{v}$ is often written as $\mathbf{u}-\mathbf{v}$.

Vectors in ${\mathbb{R}}^n$

$$\mathbf{u}=\left[\begin{array}{c}{u}_1\\ u_2\\ ...\\ u_n\end{array}\right]$$

• The vector whose entries are all zero is called the zero vector and is denoted by 0. (The number of entries in 0 will be clear from the context.)

Linear Combinations

线性组合

Linear Combinations

$$y=c_1{\mathbf{v}}_1+...+c_p{\mathbf{v}}_p$$is called a linear combination of ${\mathbf{v}}_1,...,{\mathbf{v}}_p$ with weights $c_1,...,c_p$.

The weights in a linear combination can be any real numbers

线性组合与“存在”问题的联系

• The next example connects a problem about linear combinations to the fundamental existence question studied in Sections 1.1 and 1.2.

EXAMPLE 4

Let ${\mathbf{a}}_1=\left[\begin{array}{c}1\\ -2\\ -5\end{array}\right],{\mathbf{a}}_2=\left[\begin{array}{c}2\\ 5\\ 6\end{array}\right]$, and $\mathbf{b}=\left[\begin{array}{c}7\\ 4\\ -3\end{array}\right]$. Determine whether $\mathbf{b}$ can be generated (or written) as a linear combination of ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$.

SOLUTION

• To solve this system, row reduce the augmented matrix of the system as follows:
  

$\sim$ 表示行等价

• Hence $\mathbf{b}$ is a linear combination of ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$, with weights $x_1=3$ and $x_2=2$.

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• Observe in Example 5 that the original vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and $\mathbf{b}$ are the columns of the augmented matrix that we row reduced:
  For brevity, write this matrix in a way that identifies its columns—namely,
  $$[{\mathbf{a}}_1{\mathbf{a}}_2\mathbf{b}]$$

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• The discussion above is easily modified to establish the following fundamental fact:
  

生成集 $\text{Span}$

• One of the key ideas in linear algebra is to study the set of all vectors that can be generated or written as a linear combination of a fixed set { v 1 , . . . , v p } \{\boldsymbol v_1,...,\boldsymbol v_p \} {v1​,...,vp​} of vectors.

• Asking whether a vector $\mathbf{b}$ is in S p a n { v 1 , . . . , v p } Span\{\boldsymbol v_1,...,\boldsymbol v_p \} Span{v1​,...,vp​} (生成集) amounts to asking whether the vector equation
  $$x_1{\mathbf{v}}_1+x_2{\mathbf{v}}_2+...+x_p{\mathbf{v}}_p=\mathbf{b}$$has a solution, or, equivalently, asking whether the linear system with augmented matrix $$[{\mathbf{v}}_1...{\mathbf{v}}_p\mathbf{b}]$$ has a solution.

Span { v 1 , . . . , v p } \{\boldsymbol v_1,...,\boldsymbol v_p \} {v1​,...,vp​} 一定包含 $0$

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A Geometric Description of Span { v } \{\boldsymbol v\} {v} and Span { u , v } \{\boldsymbol u,\boldsymbol v \} {u,v}

• Let $\mathbf{v}$ and $\mathbf{u}$ be a nonzero vector in ${\mathbb{R}}^3$, with $\mathbf{v}$ not a multiple of $\mathbf{u}$
  
  
• If $\mathbf{v}$ is a multiple of $\mathbf{u}$, then Span { u , v } \{\boldsymbol u,\boldsymbol v \} {u,v} can be just a line through the origin. In fact, Span { u , v } \{\boldsymbol u,\boldsymbol v \} {u,v} can also be just the origin itself.

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EXAMPLE 5

Let ${\mathbf{a}}_1=\left[\begin{array}{c}1\\ 4\\ -2\end{array}\right],{\mathbf{a}}_2=\left[\begin{array}{c}-2\\ -3\\ 7\end{array}\right]$, and $\mathbf{b}=\left[\begin{array}{c}4\\ 1\\ h\end{array}\right]$ For what value(s) of $h$ is $\mathbf{b}$ in the plane spanned by ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$?

SOLUTION

• When the linear system $[{\mathbf{a}}_1{\mathbf{a}}_2\mathbf{b}]$ is consistent.

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EXAMPLE 6

A thin triangular plate of uniform density and thickness has vertices at ${\mathbf{v}}_1=(0,1)$, ${\mathbf{v}}_2=(8,1)$, and ${\mathbf{v}}_3=(2,4)$. This “balance point” of the plate coincides with the center of mass of a system consisting of three 1-gram point masses located at the vertices of the plate. Determine how to distribute a mass of 6 g at the three vertices of the plate to move the balance point of the plate to (2,2).

SOLUTION

• The condition $w_1+w_2+w_3=6$ and the vector equation above combine to produce a system of three equations whose augmented matrix is shown below, along with a sequence of row operations:
  
• Answer: Add 3.5 g at (0, 1), add .5 g at (8, 1), and add 2 g at (2, 4).

Linear Combinations in Applications

• The final example shows how scalar multiples and linear combinations can arise when a quantity (量) such as “cost” is broken down into several categories.

EXAMPLE 7

A company manufactures two products. For $1.00 worth of product B, the company spends $.45 on materials, $.25 on labor, and $.15 on overhead (管理费用). For $1.00 worth of product C, the company spends $.40 on materials, $.30 on labor, and $.15 on overhead. Let
$$\mathbf{b}=\left[\begin{array}{c}.45\\ .25\\ .15\end{array}\right],\mathbf{c}=\left[\begin{array}{c}.40\\ .30\\ .15\end{array}\right]$$Then $\mathbf{b}$ and $\mathbf{c}$ represent the “costs per dollar of income” for the two products. Suppose the company wishes to manufacture $x_1$ dollars worth of product B and $x_2$ dollars worth of product C. Give a vector that describes the various costs the company will have (for materials, labor, and overhead).

SOLUTION
$$x_1\mathbf{b}+x_2\mathbf{c}$$