[KOCW 선형대수] 6강. 영벡터공간과 해집합

제 6강. 영벡터공간과 해집합

Vector Space

= Set of vectors in ℝn${\mathbb{}}^{}$

(1) Closed under addition ( 덧셈이 가능하다 )

V1∈𝕍,V2∈𝕍$${}_{\mathrm{}}\mathbb{}{}_{\mathrm{}}\mathbb{}$$

V1+V2∈𝕍$${}_{\mathrm{}}{}_{\mathrm{}}\mathbb{}$$

(2) Closed under scalar multiplication ( 스칼라곱이 가능하다 )

V∈𝕍,c∈ℝ$$\mathbb{}\mathbb{}$$

cV∈𝕍$$\mathbb{}$$

(3) 𝕍$\mathbb{}$ includes the zerovector( 원점 )

𝕍={0}$$\mathbb{}\mathrm{}$$

• Column space of A$$

= Set of all linear combinations of column vectors in A$$

{x|x=∑i=1nciai}$$\underset{\mathrm{}}{\overset{}{}}{}_{}{}_{}$$

A=[a1,a2,⋯,an]$${}_{\mathrm{}}{}_{\mathrm{}}{}_{}$$

Ax=b$$$$

x=⎡⎣⎢⎢⎢⎢x1 x2 ⋮ xn⎤⎦⎥⎥⎥⎥$$\mathbf{}\begin{array}{c}{}_{\mathrm{}}\\ {}_{\mathrm{}}\\ \\ {}_{}\end{array}$$

x1a1+x2a2+⋯+xnan=b${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{}{}_{}$

if, b∈ℂ(A)$\mathbb{}$ -> there are solutions ( 1개 이상 )

if, b∉ℂ(A)$\mathbb{}$ -> no solution

• Span : 벡터들로 만들 수 있는 모든 Linear Combinations 집합 Whole space is constructed by linear combinations

Null space of A ( N(A)$$ )

= Set of vectors such that Ax=0$\mathrm{}$

There are infinite solutions

They form a vector space - null space of A

• The null space of a matrix A consists of all vectors x such that Ax=0$\mathrm{}$, N(A)$$ -> subspace

(i) Closed under addition

Ax=0$\mathrm{}$ and Ax′=0${}^{}\mathrm{}$ -> A(x+x′)=0${}^{}\mathrm{}$, closed under addition

(ii) Closed under scalar multiplication

Ax=0$\mathrm{}$ -> Acx=c0=0$\mathrm{}\mathrm{}$, closed under scalar multiplication (for any C)

ex) (1)

A=⎡⎣⎢⎢152044⎤⎦⎥⎥[uv]=⎡⎣⎢⎢000⎤⎦⎥⎥$$\mathbf{}\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}$$

-> homogeneous equation!  Ax=0$\mathrm{}$

N(A)=(0,0)$\mathrm{}\mathrm{}$

(2)

B=⎡⎣⎢⎢152044196⎤⎦⎥⎥⎡⎣⎢⎢uvw⎤⎦⎥⎥=⎡⎣⎢⎢000⎤⎦⎥⎥$$\mathbf{}\begin{array}{ccc}\mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \\ \end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}$$

-> same ℂ(A),ℂ(B)$\mathbb{}\mathbb{}$

--> N(B)$$ is in the line (c,c,-c) (1,1,-1)

we will find N(A),ℂ(A)$\mathbb{}$

Solving Ax=0$\mathrm{}$(null space) & Ax=b$$

미지수 개수(unknowns) > 식의 개수(eqns)

=> 해가 아예 없거나(평행), 무수히 많다.

• For an invertible matrix A$$, the null space N(A)$$ contains only x=0$\mathrm{}$, The column space is the whole space

• When the null space contains more than the zero vector column space contains less than all vectors

• b$$ complete solution Axp=b${}_{}$ and Axn=0${}_{}\mathrm{}$ -> A(xp+xn)=b${}_{}{}_{}$

ex)

 

[1212][yz]=[b1b2]$$\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \end{array}\begin{array}{c}{}_{\mathrm{}}\\ {}_{\mathrm{}}\end{array}$$

(i) b2≠2b1${}_{\mathrm{}}\mathrm{}{}_{\mathrm{}}$

-> No-Solution

(ii) b2=2b1${}_{\mathrm{}}\mathrm{}{}_{\mathrm{}}$

-> infinitely many solutions

xp${}_{}$ = (1,1)

=> 

[1212][yz]=[24]$$\begin{array}{cc}\mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\end{array}$$

xp${}_{}$가 1,1이면 b_1과 b_2는 2,4가 된다.

xn${}_{}$ = (-1,1) or (-C, C)

xp+xn=[11]+C[−11]=[1−C1+C]$${\mathbf{}}_{\mathbf{}}\mathbf{}{\mathbf{}}_{\mathbf{}}\begin{array}{c}\mathrm{}\\ \mathrm{}\end{array}\mathbf{}\begin{array}{c}\mathrm{}\\ \mathrm{}\end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\end{array}$$

x=xp+xn${}_{}{}_{}$에 대한 해결책들을 line으로 나타낸다.

Echelon Form U and Row Reduced Form R for rectangular mat

-> the simplest matrix that elimination can give it

ex)

A=⎡⎣⎢⎢12−136−3393274⎤⎦⎥⎥−>A=⎡⎣⎢⎢100300336236⎤⎦⎥⎥$$\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}$$

Pivot 2 가 0이라 column 2의 Pivot이 존재하지 않는다.

A=⎡⎣⎢⎢100300330230⎤⎦⎥⎥$$\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}$$

3번식 - 2* 2번식

• Pivot 위치에 따라 Upper triangular이 된다. == 이것이 "Echelon form U" 이다. 그리고 2번식을 3으로 나눈다

A=⎡⎣⎢⎢100300310210⎤⎦⎥⎥$$\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}$$

모든 Pivot들이 1이 되었다.

A=⎡⎣⎢⎢100300010−110⎤⎦⎥⎥$$\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}$$

위의 식은 1번식 - 3* 2번식의 결과이다.

이를 " Row Reduced form R " 형태이다. ( 모든 Pivot이 1 )

이는 Pivot위의 성분들을 다 0으로 만들어줄 때 쓰인다.

Ax=0$\mathrm{}$ -> Ux=0$\mathrm{}$ and Rx=0$\mathrm{}$ ( same solution )

Pivot Variables and Free Variables

for solving Ax=0$\mathrm{}$, the Pivots are " important "

Ax=⎡⎣⎢⎢100300010−110⎤⎦⎥⎥⎡⎣⎢⎢⎢⎢uvwy⎤⎦⎥⎥⎥⎥=⎡⎣⎢⎢000⎤⎦⎥⎥$$\mathbf{}\mathbf{}\begin{array}{cccc}\mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\\ \mathrm{}& \mathrm{}& \mathrm{}& \mathrm{}\end{array}\begin{array}{c}\\ \\ \\ \end{array}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}$$

빨간색으로 표시된 성분들은 " Special solution ( reverse sign ) " 이다.

Pivots는 첫 번째, 세 번째 열에 위치해 있다.

• Pivot Variables with pivot columns = u,w$$
• Free Variables without pivot columns = v,y$$

solve = u+3v−y=0$\mathrm{}\mathrm{}$ and w+y=0$\mathrm{}$

u=−3v+y$$\mathrm{}$$

w=−y$$$$

x=⎡⎣⎢⎢⎢⎢uvwy⎤⎦⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢−3v+yv−yy⎤⎦⎥⎥⎥⎥=v⎡⎣⎢⎢⎢⎢−3100⎤⎦⎥⎥⎥⎥+y⎡⎣⎢⎢⎢⎢10−11⎤⎦⎥⎥⎥⎥$$\mathbf{}\begin{array}{c}\\ \\ \\ \end{array}\begin{array}{c}\mathrm{}\\ \\ \\ \end{array}\mathbf{}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}\mathbf{}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}$$

• Special Solution : (-3,1,0,0) , (1,0,-1,1) <= (v=1, y=0) and (v=0, y=1)

=> Null space contains all combinations of special solutions using free variables

v,y 와 같은 개수가 null space의 diemension이 됨. 모양새가 4차원 같지만 사실은 2차원이다.

(미지수 = 4 , 표현 가능 벡터 = 2)

원래 

⎡⎣⎢⎢⎢⎢uvwy⎤⎦⎥⎥⎥⎥inℝ4$$\begin{array}{c}\\ \\ \\ \end{array}{\mathbb{}}^{\mathrm{}}$$

사실 Dim(N(A)$$) = 2

나머지 2차원은 row space( 행 벡터 )