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!
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