Week 2 : Introduction of Linear Algebra

연립방정식

3x+ 2y = 1  --> 식 하나하나가 어떤 현상을 설명하는 단서

식이 여러개 있는 것 : system of linear algebraic equations

 

Search engine을 만들때 선형대수학이 근간이 된다

- 선형대수학을 이용해서 행렬을 변형을 시킨다. 

- SVD(Singular Value Decomposition)라는 format으로 변형됨

- 키워드를 쳤을 때, 키워드를 끌어내는 조작을 하는데 사용

- SVD : 행렬을 특정한 구조로 분해하는 방식으로, 신호 처리와 통계학 등의 분야에서 자주 사용

 

CPS(Cyber Physical System)

- CPS의 근간이 되는것도 Matrix

- 디지털 트윈은 현실세계의 기계나 장비, 사물 등을 컴퓨터 속 가상세계에 구현한 것을 말한다

- 현실에 있던 요소들의 number들을 matrix을 통해서 mapping을 하는 작업

- 행렬의 시작은 (2x2, 3x3..처럼)간단하지만 실제 계산으로 들어가면 10000x10000정도로 매우 큰 정보를 계산해야함

 

Review of matrices

- Systems of linear algebraic equations AX=B

----> Finding solution(s) by row operations(=Gaussian elimination)

- Inverse of a square matrix A

----> Finding inverse by row operations

- Determinant of a square matrix A 

----> Calculating determinant by row operations

 

What is a matrix?

- An MxN matrix is an array of MN numbers enclosed within a pair of brackets and arranged in M rows and N columns. 

- The numbers making up a matrix are referred to as “elements” of the matrix.

- In a square matrix, the number of rows equals the number of columns.

- We use bold capital letters such as A, B, P and Q to denote matrices

 

Equality of matrices

- element가 같으면 된다

- 당연히 matrix의 크기도 같아야 함(=same order)

- A and B are said to be equal, that is, A = B, if aij = bij

 

Addition of matrices

- If A+B = C then C=(cij) is MxN and cij = aij + bij

 

Multiplication of a number to a matrix

- If A = (aij) and c is a number then cA = (caij) and cA has the same order as A

 

Product of matrices

- Condition : Let A=(aij) and B=(bij) be MxN and PxQ matrices respectively. If N=P, we can form the product matrix AB

- Multiplication of matrices is not commutative, that is, even if AB and BA can be formed, AB may or may not be equal to BA.

 

Identity matrices

- An NxN matrix (cij) such that c11 = c22 = c33=…= cNN =1 and cij = 0 for i not equal to j is called an identity matrix

- Let I be an identity matrix and A be any matrix.

- If the product IA can be formed then IA = A. Similarly, if the product AI can be formed then AI = A.

 

Transpose of a matrix

- If A is an M×N matrix then the transpose of A is the N×M matrix

- “The i-th column of the transpose of A is the i-th row of A.

 

What’s a linear algebraic equation?

- An example of a linear algebraic equation in one unknown x is : 2x + 1 = 0

- A linear algebraic equation in two unknowns x and y is an equation of the form : 3x - y = 9

---> Generalization

A linear algebraic equation in N unknowns x1 , x2 , …, xN-1 and xN is an equation of the form:

c1x1 + c2x2+ ... + cNxN = dN

 

- Many problems in engineering and physical sciences are formulated in terms of a system of linear algebraic equations

----> 찾아야하는 unknown이 N개로 너무 많아서 어렵다..--> 이걸 찾는게 Linear Algebra!!

- 사실 우리 실생활에서 일어나는 일들은 비선형인것들이 훨씬 많다. ------> 이런 것들을 잘게잘게 나누면 선형화시킬 수 있다!! --> 선형화 시킨 후 해석을 할때는 컴퓨터를 이용한다!

 

SOLUTION SYSTEM

 

1. Inconsistent System

It is possible that a system of linear algebraic equations has no solution.!

- x + y = 11, x + y = 34

 

2. Consistent System

It is possible that a consistent system has more than one solutions.!

- x + y = 10, 2x + 2y  = 20

- The system really contains only one linear algebraic equation in 2 unknowns!

- So we can find infinitely many solutions for the system.

 

-  If a consistent system of linear algebraic equations has only one solution, we say the system has a unique solution.

 

Given a system of linear algebraic equations AX = B, how do we know whether it is consistent or not? If it is consistent, how do we find all its solutions?

- Reduce the system AX = B to a simpler but equivalent system UX = C(U == upper triangular matrix)

- <Define> : AX = B and UX = C are equivalent if they have exactly the same solution(s).

- If we can work out the solution(s) of UX = C, we would have solved AX = B.(By Definition)

- If the square matrix U is an upper triangular matrix, the system UX = C would be simpler enough for us to work out its solution(s) (if any).

 

What is an upper triangular matrix? <----> Lower triangular matrix

- Diagonal element 기준, 위쪽으로는 수가 있고 아래쪽으로는 0만 있는 matrix!

 

Solve

-  row operation(= Gaussian elimination)을 통해 upper triangular matrix로 만든 후 삼각형의 아래쪽 unknown부터 위쪽 unknown까지 차례대로 대입하여 푼다.!(Backward Substitution)

 

There are 2 types of legitimate row operations.

- Ri <----> Rj Interchange i-th and j-th rows. (바꿔도 solution은 변함 없으므로 성립)

- Ri ---> aRi + bRj  Use row j to change row i to become  aRi + bRj .(The Constant 'a' is not allowed to be zero!)

---> if a = 0, b = 1 ; Ri를 Rj로 바꾸면 정보(식)가 하나 없어지는 것이다!