1.3 Vector Equations

조범희 선생님의 선형대수학개론 강의를 듣고 공부하며 정리한 내용입니다.
정말 좋은 강의힙니다. 강추합니다. 링크

Vectors in $\mathbb{R}^2$

$\mathbb{R}^2$ = real space (실수 2차원 공간)

$\bold{u} = \vec{u} = \begin{bmatrix}3\\-1\end{bmatrix}$ $\bold{v} = \vec{v} = \begin{bmatrix}0.3\\0.1\end{bmatrix}$ $\bold{w} = \vec{w} = \begin{bmatrix}w_1\\w_2\end{bmatrix}$

 

Vector Summation

$\bold{u} + \bold{v} = \begin{bmatrix}3\\-1\end{bmatrix} +\begin{bmatrix}0.3\\0.1\end{bmatrix} = \begin{bmatrix}3.3\\-0.9\end{bmatrix}$

 

Scalar multiplication

$c = 5,\ \bold{u} = \begin{bmatrix}3\\-1\end{bmatrix}\ \ c * u = 5\begin{bmatrix}3\\-1\end{bmatrix} = \begin{bmatrix}5*3\\5*(-1)\end{bmatrix} = \begin{bmatrix}15\\-5\end{bmatrix}$

 

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

$\bold{u} = \begin{bmatrix}2\\2\end{bmatrix},\ \bold{v} = \begin{bmatrix}-6\\1\end{bmatrix},\ \bold{u+v} = \begin{bmatrix}-4\\3\end{bmatrix}$

$\bold{u} = \begin{bmatrix}3\\-1\end{bmatrix},\ 2\bold{u},\ -\dfrac{2}{3}\bold{u}$

 

어떤 vector u 에 임의의 scalar 값을 곱하면, 그 vector cu 는 vector u 와 동일선상에 있게 된다.

따라서, 이때의 c를 실수 전체로 가정하면, vector cu 가 그리는 발자취는 직선과 같게 된다.

이를 set of all multiple of u 이라고 한다.

 

Vectors in $\mathbb{R}^3$

$\bold{a} = \begin{bmatrix}1\\5\\4\end{bmatrix}$

 

Vectors in $\mathbb{R}^n$

$\bold{u} = \begin{bmatrix} u_1\\u_2\\\vdots\\u_n \end{bmatrix} = (u_1, u_2, \cdots, u_n)$

 

Algebraic properties of $\mathbb{R}^n$

$\bold{u}\quad \bold{v}\quad \bold{w}\quad in\ \ \mathbb{R}^n$

$c\quad d\quad is\ scalar$

properties

$i)\quad \bold{u}+\bold{v} = \bold{v} + \bold{u}$

$ii)\quad (\bold{u}+\bold{v}) + \bold{w} = \bold{u} + (\bold{v} + \bold{w})$

$iii)\quad \bold{u}+\bold{0} = \bold{0} + \bold{u} = \bold{u} \quad ,(\bold{0} = \vec{0})$

$iv)\quad \bold{u}+(-\bold{u}) = -\bold{u} + \bold{u} = \bold{0} \quad ,(\bold{0} = \vec{0})$

$v)\quad c(\bold{u}+\bold{v}) = c\bold{u}+c\bold{v}$

$vi)\quad (c+d)\bold{u} = c\bold{u}+d\bold{u}$

$vii)\quad c(d\bold{u}) = (cd)\bold{u}$

$viii)\quad 1\bold{u} = \bold{u}$

 

Linear Combinations

$\bold{v_1},\ \bold{v_2},\ \cdots,\ \bold{v_p}\quad in\ \ \mathbb{R}^n$

$c_1,\ c_2,\ \cdots,\ c_p\quad is\ scalar$

$y = c_1\bold{v_1} + c_2\bold{v_2} +\cdots+c_p\bold{v_p}$

$y$ is linear combination of $\bold{v_1},\ \bold{v_2},\ \cdots,\ \bold{v_p}$ with weights $c_1,\ c_2,\ \cdots,\ c_p$

 

Example 1. can b be generated as a linear combination of a1 and a2 ?

$\bold{a_1} = \begin{bmatrix} 1\\-2\\-5 \end{bmatrix}$

$\bold{a_2} = \begin{bmatrix} 2\\5\\6 \end{bmatrix}$

$\bold{b} = \begin{bmatrix} 7\\4\\-3 \end{bmatrix}$

 
Solution.

$x_1\bold{a_1} + x_2\bold{a_2} = \bold{b}$

 
$x_1\begin{bmatrix} 1\\-2\\-5 \end{bmatrix} + x_2\begin{bmatrix} 2\\5\\6 \end{bmatrix} = \begin{bmatrix} x_1+2x_2\\-2x_1+5x_ 2\\-5x_1+6x_2 \end{bmatrix} = \begin{bmatrix} 7\\4\\-3 \end{bmatrix}$

 
$\begin{bmatrix} 1 & 2 & 7\\-2 & 5 & 4\\-5 & 6 & -3 \end{bmatrix} \sim\begin{bmatrix} 1 & 2 & 7\\0 & 9 & 18\\0 & 16 & 32 \end{bmatrix} \sim\begin{bmatrix} 1 & 2 & 7\\0 & 1 & 2\\0 & 16 & 32 \end{bmatrix} \sim\begin{bmatrix} 1 & 2 & 7\\0 & 1 & 2\\0 & 0 & 0 \end{bmatrix} \sim\begin{bmatrix} 1 & 0 & 3\\0 & 1 & 2\\0 & 0 & 0 \end{bmatrix}$

 
$\begin{cases} x_1 = 3\\ x_1 = 2 \end{cases}$

 

can b be generated as a linear combination of a1 and a2 ?

 

= Does the follwing augmented matrix have a solution?

$\begin{bmatrix} \bold{a_1}\quad \bold{a_2}\quad \bold{b}\end{bmatrix}$

 

so,

A vector equation

$x_1\bold{a_1} + x_2\bold{a_2} + \cdots + x_n\bold{a_n} = \bold{b}$

 

has the same solution set as the linear system whose augmented matrix is

$\begin{bmatrix} \bold{a_1}\ \bold{a_2}\ \cdots\ \bold{a_n}\ \bold{b}\end{bmatrix}$

 

Span $\begin{Bmatrix}\bold{v_1}, & \cdots, & \bold{v_p}\end{Bmatrix}$

 
is the collection of all vectors that can be written in the form

$c_1\bold{v_1} + c_2\bold{v_2} +\cdots+c_p\bold{v_p}$

 

Is a vector b in span $\begin{Bmatrix} \bold{v_1}, & \cdots, & \bold{v_p} \end{Bmatrix}$ ?

 

= Does the follwing vector equation have a solution?

$x_1\bold{v_1} + x_2\bold{v_2} + \cdots + x_n\bold{v_p} = \bold{b}$

 

= Does the follwing augmented matrix have a solution?

$\begin{bmatrix} \bold{v_1}\quad \cdots\quad \bold{v_p}\quad \bold{b}\end{bmatrix}$

 

따라서, 'Is a vector b in span $\begin{Bmatrix} \bold{v_1}, & \cdots, & \bold{v_p} \end{Bmatrix}$ ?' 꼴로 질문이 많이 주어진다

 

Geometric descriptions of Span $\begin{Bmatrix}\bold{v}\end{Bmatrix}$ and Span $\begin{Bmatrix}\bold{u}, & \bold{v}\end{Bmatrix}$ in $\mathbb{R}^3$

(u랑 v는 scalar multiplication 으로 표현되지 않는 서로 다른 vector 이다)

span을 이용해서 표현하면 공간에서의 직선과 평면을 간단하게 표현할 수 있다.

 

Example 2.

$\bold{a_1} = \begin{bmatrix} 1\\-2\\3 \end{bmatrix}$ $\bold{a_2} = \begin{bmatrix} 5\\-13\\3 \end{bmatrix}$ $\bold{b} = \begin{bmatrix} -3\\8\\1 \end{bmatrix}$

 

Solution.

$\begin{bmatrix} 1 & 5 & -3\\-2 & -13 & 8\\3 & 3 & 1 \end{bmatrix} \sim\begin{bmatrix} 1 & 5 & -3\\0 & -3 & 2\\0 & -18 & 10 \end{bmatrix} \sim\begin{bmatrix} 1 & 5 & -3\\0 & -3 & 2\\0 & 0 & -2 \end{bmatrix}$

by Theorem 2 (Existence and Uniqueness Theorem), this linear system has no solution (inconsistent)