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선형대수학, ? (if det(A)=0 adj(A)= speicfic form?, equivalent system =…
선형대수학
1
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
u•v=(u^t)v
Matrix multiplication
matrix multiplication is not commutative
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
scalar matrix
identity matrix
square matrix
triangular
upper triangular
lower triangular
symmetric
skew symmetric
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
a unique solution
nontrivial solution
infinitely many solutions
unique solution
infinitely many solutions
equivalent system
2
5.Inverse Matrix
invertible - nonsingular
AB=BA=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
Gauss-Jordan reduction
contradiction
parameter
row equivalent
row operations
finite number?
8.Determinant
definition
permutation
Inversion
ex) 132=1
even/odd
number of inversion
number of permutation
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
?
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?
equivalent system = infinitely many solutions 역도 가능?
A,B square matrix AB=In BA is not in
theorem 3.1
how to get solution in linear algebra
1.row operation
echelon form
reduced row operation
determinant
unique solution
A inverse
crammer rule
REF (row echelon form)
RREF(reduced row echelon form)
