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선형대수학, ? (if det(A)=0 adj(A)= speicfic form?, equivalent system =…

- - - - rectangular array of numbers
    - - column ㅣ
    - - nx1 matrix
    - - m=n
      - main diagonal
    - - the set of all mxn matrices
    - - matrix addition
      - scalar multiplication
      - linear combination
        
        coefficient
      - transpose
      - multiplication
        
        dot product
        
        inner product
        
        u•c=R
        
        x•x>0
        
        u•v=(u^t)v
        
        Matrix multiplication
        
        matrix multiplication is not commutative
        
        matrix-vector product
      - Augmented matrix
  - - - commutative low
        
        A+B=B+A
      - Associative low
        
        A+(B+C)=(A+B)+C
      - zero matrix
        
        0
    - - associative properties
        
        A(BC)=(AB)C
      - distributive property
        
        (A+B)=AC+BC
        
        C(A+B)=CA+CB
      - is not commutative
      - Ax=0 c1x1+c2x2 is also solution
    - - AB=0 , A=0 or B=0
      - AB=AC then B=C
      - AB=BC
  - - - scalar matrix
      - identity matrix
    - - upper triangular
      - lower triangular
    - - skew symmetric
  - - - linear equation of n equations in m unknowns
    - - inconsistent
        
        no solution
      - consistent
        
        has solution
        
        homogeneous system
        
        trivial solution
        
        a unique solution
        
        nontrivial solution
        
        infinitely many solutions
        
        unique solution
        
        infinitely many solutions
        
        equivalent system
- - - - uniqueness
        
        B=~~=C
  - - - counterclockwise rotation
  - - - condition of echelon form
        
        1.all zero
        
        2.leading one
        
        3.to the right and below
        
        4.column all zero
    - - Gauss-Jordan reduction
      - contradiction
      - parameter
    - - row operations
      - finite number?
  - - - permutation
        
        Inversion
        
        ex) 132=1
        
        even/odd
        
        number of inversion
        
        number of permutation
        
        n!
    - - tranpose
      - all 0 row(column)
      - interchange row(column)
      - equal row(column)
      - scalar
      - row operation
      - triangular
      - I
      - inverse
      - product
    - - Minor
      - I≠K 0

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선형대수학

?

1

1.linear equations

how to read

linear equation of n equations in m unknowns

solutions

inconsistent

no solution

consistent

has solution

homogeneous system

trivial solution

nontrivial solution

?

equivalent system = infinitely many solutions 역도 가능?

unique solution

infinitely many solutions

equivalent system

a unique solution

infinitely many solutions

2.Matrix

definition

rectangular array of numbers

row ㅡ

column ㅣ

vector

nx1 matrix

square matrix

m=n

main diagonal

the set of all mxn matrices

Matrix operation

matrix addition

scalar multiplication

linear combination

coefficient

transpose

multiplication

dot product

inner product

u•c=R

x•x>0

Matrix multiplication

matrix multiplication is not commutative

u•v=(u^t)v

matrix-vector product

Augmented matrix

3.properties matrix operation

matrix addition

commutative low

A+B=B+A

Associative low

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

zero matrix

0

matrix multiplication

associative properties

A(BC)=(AB)C

distributive property

(A+B)=AC+BC

C(A+B)=CA+CB

is not commutative

Ax=0 c1x1+c2x2 is also solution

transpose

disprove

AB=0 , A=0 or B=0

AB=AC then B=C

AB=BC

4.special types of Matrices

diagonal matrix

square matrix

scalar matrix

identity matrix

triangular

upper triangular

lower triangular

symmetric

skew symmetric

2

5.Inverse Matrix

invertible - nonsingular

AB=BA=In

A,B square matrix AB=In BA is not in

properties

uniqueness

B=~~=C

unique solution

6.vector(1)

Reflection

projection

Dilation and contraction

Rotation

counterclockwise rotation

7.Echelon Form

definition

condition of echelon form

1.all zero

2.leading one

3.to the right and below

4.column all zero

elementary row operations

row equivalent

row operations

finite number?

REF (row echelon form)

RREF(reduced row echelon form)

Gauss-Jordan reduction

contradiction

parameter

8.Determinant

definition

permutation

Inversion

even/odd

number of inversion

number of permutation

ex) 132=1

if det(A)=0 adj(A)= speicfic form?

ex (1 2 3, 0 2 2 , 1 4 5) adj(A)= (2 2 -2 , 2 2 -2 , -2 -2 0 )

A*B=0 B is a unique matrix?

theorem 3.1

how to get solution in linear algebra

1.row operation

determinant

unique solution

A inverse

crammer rule

echelon form

reduced row operation

n!

properties

tranpose

all 0 row(column)

interchange row(column)

equal row(column)

scalar

row operation

triangular

I

inverse

product

Cofactor

Minor

I≠K 0