[KOCW 선형대수] 5강. 벡터공간과 열벡터공간

제 5강. 벡터공간과 열벡터공간

미지수 = 방정식 => Unique or No-Solution

미지수 > 방정식 => Infinitely or Many Solutions

Vector Spaces and Subspaces

• Space : Set closed under addition & scalar multiplication

           원소갖는 집합, 덧셈 닫힘 / scala 곱 닫힘 ( 닫힘 = 포함, 가능 )

for any vectors  X,Y∈ℝn${\mathbb{}}^{}$

for any scalar C∈ℝn${\mathbb{}}^{}$

X,Y∈𝕍$\mathbb{}$

{X+Y∈𝕍CX∈𝕍$$\begin{array}{l}\mathbb{}\\ \mathbb{}\end{array}$$

C1X+C2Y∈𝕍${}_{\mathrm{}}{}_{\mathrm{}}\mathbb{}$ (V는 벡터공간)

A real vector space ℝn${\mathbb{}}^{}$(n-dimmensional)

(1) x+y=y+x$$

(2) $ x + ( y + z ) = ( x + y ) + z

(3) There is a unique "Zero vector", such that x+0=0+x=x$\mathrm{}\mathrm{}$

(4) For each x$$, there is a unique vector −x$$, such that −x+x=0$\mathrm{}$ ( 역원:unique하다)

(5) 1•x=x$\mathrm{}$

(6) $(C_1C_2)x = C_1(C_2x)

(7) C(x+y)=Cx+Cy$$

(8) (C1+C2)x=C1x+C2x${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}$

{x,y:VectorC,C1,C2:Scalar$$\begin{array}{l}\\ {}_{\mathrm{}}{}_{\mathrm{}}\end{array}$$

• x$$ and y$$ should be cloased under addition and scalar multiplication => " Space " 

ex) m$$ x n$$ matrix ∈ℝmn${\mathbb{}}^{}$

2x2 matrix 이면  ∈ℝmn${\mathbb{}}^{}$

• Subspace : non-empty subset that satisfies the requirements for a vector space 

(i) x+y∈subspace$$ , for ∀x,y∈subspace$\mathrm{}$

(ii) cx∈subspace$$ , for ∀x∈subspace,∀c$\mathrm{}\mathrm{}$

=> x$$ and y$$ should be closed under vector addition and scalar multiplications in the subspace

• different from subset 

ex) (i) ℝ3${\mathbb{}}^{\mathrm{}}$ vectors on a plane(평면) which passes through the origin(원점)

(ii) the origin => smallest subspace

(iii) ℝ2${\mathbb{}}^{\mathrm{}}$ vectors on a line which passes through the origin.

(iv) lower triangular matrices

c1L1+c2L2 in𝕃${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}\mathbb{}$

Lnxn⊂Anxn∈ℝn2${}_{}{}_{}{\mathbb{}}^{{}^{\mathrm{}}}$

• The Column Space of A

• Column space : contains all linear combinations of the columns of matrix A

Ax=b$$ (mxn) =>

⎡⎣⎢⎢1 a1 11a21⋯⋯⋯1an1⎤⎦⎥⎥⎡⎣⎢⎢⎢⎢x1 x2 ⋮ xn⎤⎦⎥⎥⎥⎥=⎡⎣⎢⎢⎢⎢b1 b2 ⋮ bn⎤⎦⎥⎥⎥⎥$$\begin{array}{cccc}\mathrm{}& \mathrm{}& & \mathrm{}\\ {}_{\mathrm{}}& {}_{\mathrm{}}& & {}_{}\\ \mathrm{}& \mathrm{}& & \mathrm{}\end{array}\begin{array}{c}{}_{\mathrm{}}\\ {}_{\mathrm{}}\\ \\ {}_{}\end{array}\begin{array}{c}{}_{\mathrm{}}\\ {}_{\mathrm{}}\\ \\ {}_{}\end{array}$$

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

-> linear combinations of column vectors equals b$$

• Ax=b$$는 Solvable할 수 있다. 만약 b$$가 A$$의 Column의 결합으로 표현될 수 있다면 가능하다. 그러면 b$$는 column space에 있는 것이다. 

u⎡⎣⎢⎢1 5 2⎤⎦⎥⎥+v⎡⎣⎢⎢0 4 4⎤⎦⎥⎥=⎡⎣⎢⎢b1 b2 b3⎤⎦⎥⎥$$\mathbf{}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}\mathbf{}\begin{array}{c}\mathrm{}\\ \mathrm{}\\ \mathrm{}\end{array}\begin{array}{c}{}_{\mathrm{}}\\ {}_{\mathrm{}}\\ {}_{\mathrm{}}\end{array}$$

평면 위에 존재한다면 -> 해가 존재하고

평면 밖에 있다면 -> 해가 존재하지 않는다.

• Column Space 

if b1,b2∈ℂ(A)${}_{\mathrm{}}{}_{\mathrm{}}\mathbb{}$ A라는 column space의 원소이다.

Ax1=b1${}_{\mathrm{}}{}_{\mathrm{}}$

Ax2=b2${}_{\mathrm{}}{}_{\mathrm{}}$

b1+b2=b${}_{\mathrm{}}{}_{\mathrm{}}$

Ax1+Ax2=A(x1+x2)=b${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}$

• Column space ℂ(A)$\mathbb{}$

(i) if b$$ and b′${}^{}$ lie in the column space -> $A(x+x') = b + b'

-> 덧셈에 대해 닫혀있다.(가능하다)

(ii) if b$$ lies in the column space -> A(cx)=cb$$

-> 스칼라 곱에 대해 닫혀있다.

ex) (i) for any non-singular 5x5 matrix -> ℝ5${\mathbb{}}^{\mathrm{}}$ (column) space

-> (5 pivots in Gauss Elimination )

(ii) Zero matrix -> 0 - dimensional space