[Linear Algebra] #01 Matrices, Vectors, Determinants Linear Systems

1. 기본 용어

 

Linear : 선형의

Algebra : 대수, 연산, 계산

Matix : 행렬

Vector : column & row 벡터

Determinant : |[ - ]| = 0 → 행렬식

Linear System : 입력과 출력이 존재하는 선형 시스템(ex. 함수)

 

entries : elements of matrix

rows : 행. horizontal lines of entries

columns : 열. vertical lines of entries

square matrix : 정사각 형태 matrix

rectangular matrix : 직사각 형태 matrix

 

 

 

2. Addition and scalar Multiplication

 

· scalar : 2, 3, x, y

· vector : $\left[\begin{array} {rrr} 1 \\ 4 \end{array}\right] $

· matrix : $\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] $

 

 

 

3. General concepts and Notations

 

m x n matrix

 

· if, m = n, A is m x n square matrix.

· if, m ≠ n, m ≠ 0, n ≠ 0, A is rectangular matrix.

 

cf) main diagonals : 대각선 성분들 (main diagonal entries)

 

 

 

4. Addition and scalar Multiplication of Matrices and vectors

 

Def. Equality of Matrices

A = B, A 와 B 의 dimension, size 가 같고 모든 A, B 의 entries 들이 서로 같다.

 

Def. Addition of Matrices

A + B. 같은 자리끼리 더함.

 

Def. Scalar Multiplication (Multiplication by a number)

cA.

 

 

 

5. Rules for Matrix Addition and scalar Multiplication

 

· Matrix Addition

 

(a) A + B = B + A 

→ commutative property(교환법칙)

 

(b) (A + B) + C = A + (B + C)

→ associative property(결합법칙)

 

(c) A + 0 = A                                                       

cf) 0 → zero matrix(vector) / Identity : [0, 0, 0]

 

(d) A + ( -A ) = 0

 

 

· Scalar Multiplication

 

(a) c(A + B) = cA + cB

 

(b) (c + k)A = cA + kA

→ distributive property(분배법칙)

 

(c) c(kA) = (ck)A

 

(d) 1A = A

 

 

 

6. Matrix Multiplication

 

Def. Multiplication of a matrix by a matrix

product C = AB,  m x n matrix A = [Ajk]

                       r x p matrix B = [Bjk]

 

         if n = r,  C is m x p matrix.

 

 

· Rules for Multiplication

 

(a) (kA)B = k(AB) = A(kB)

 

(b) A(BC) = (AB)C = ABC

 

(c) (A + B) = AC + BC

 

(d) C(A + B) = CA + CB

 

 

 

7. Augmented Matrix (tilde A)

 

Augmented Matrix

 

· 특성

(1) Ax = b 라는 선형 시스템의 모든 특성을 나타낸다.

(2) x1, x2, x3 즉, input vector를 구할 때 사용된다.