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데코수학/ 선형대수학/ 행렬 1
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데코수학/ 선형대수학/ 행렬 1
(유튜브 동영상인데 현재는 삭제되어서 내용만 남김)
개념
•
행렬의 정의
◦
행렬이란 벡터공간 위에 있는 선형 함수
◦
행렬이란 숫자들의 2차원 배열
◦
i
=
1
,
2
,
.
.
.
,
m
,
j
=
1
,
2
,
.
.
.
,
n
,
a
i
j
∈
F
i = 1, 2, ... , m, j = 1, 2, ... , n, a_{ij} \in \mathbb{F}
i
=
1
,
2
,
...
,
m
,
j
=
1
,
2
,
...
,
n
,
a
ij
∈
F
일 때
▪
A
=
(
a
11
a
12
.
.
.
a
1
n
a
21
a
22
.
.
.
a
2
n
.
.
.
a
m
1
a
m
2
.
.
.
a
m
n
)
A = \left( \begin{array}{rrrr} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ ... \\ a_{m1} & a_{m2} & ... & a_{mn} \end{array} \right)
A
=
a
11
a
21
...
a
m
1
a
12
a
22
a
m
2
...
...
...
a
1
n
a
2
n
a
mn
를
F
\mathbb{F}
F
위의
m
×
n
m \times n
m
×
n
행렬이라 한다.
•
행렬 표기법
◦
A
=
a
i
j
A = a_{ij}
A
=
a
ij
◦
M
m
,
n
(
F
)
M_{m, n}(\mathbb{F})
M
m
,
n
(
F
)
:
F
\mathbb{F}
F
위의 모든
m
×
n
m \times n
m
×
n
행렬의 집합
◦
[
A
]
i
[A]_{i}
[
A
]
i
: A의 i번째 행 (
1
×
n
1 \times n
1
×
n
벡터)
◦
[
A
]
j
[A]^{j}
[
A
]
j
: A의 j번째 열 (
m
×
1
m \times 1
m
×
1
벡터)
•
A
=
(
a
i
j
)
,
B
=
(
b
i
j
)
∈
M
m
,
n
(
F
)
A = (a_{ij}), B = (b_{ij}) \in M_{m, n}(\mathbb{F})
A
=
(
a
ij
)
,
B
=
(
b
ij
)
∈
M
m
,
n
(
F
)
일 때
◦
A
=
B
⇔
∀
i
,
j
,
a
i
j
=
b
i
j
A = B \Leftrightarrow \forall_{i,j}, a_{ij} = b_{ij}
A
=
B
⇔
∀
i
,
j
,
a
ij
=
b
ij
◦
A
+
B
:
=
(
a
i
j
+
b
i
j
)
A + B := (a_{ij} + b_{ij})
A
+
B
:=
(
a
ij
+
b
ij
)
◦
c
A
:
=
(
c
⋅
a
i
j
)
c A := (c \cdot a_{ij})
c
A
:=
(
c
⋅
a
ij
)
•
A
=
(
a
i
j
)
∈
M
m
,
n
,
B
=
(
b
i
j
)
∈
M
n
,
l
A = (a_{ij}) \in M_{m, n}, B = (b_{ij}) \in M_{n, l}
A
=
(
a
ij
)
∈
M
m
,
n
,
B
=
(
b
ij
)
∈
M
n
,
l
일 때
◦
A
B
:
=
(
∑
x
=
1
n
a
i
x
b
x
j
)
∈
M
m
,
l
AB := (\sum_{x=1}^{n} a_{ix} b_{xj}) \in M_{m, l}
A
B
:=
(
∑
x
=
1
n
a
i
x
b
x
j
)
∈
M
m
,
l
▪
a
b
i
j
=
[
A
]
i
⋅
[
B
]
j
ab_{ij} = [A]_{i} \cdot [B]^{j}
a
b
ij
=
[
A
]
i
⋅
[
B
]
j
•
0
m
,
n
=
(
0
)
0_{m, n} = (0)
0
m
,
n
=
(
0
)
•
I
n
=
(
δ
i
j
)
I_{n} = (\delta_{ij})
I
n
=
(
δ
ij
)
◦
δ
i
j
⇒
1
(
i
=
j
)
,
0
(
i
≠
j
)
\delta_{ij} \Rightarrow 1 (i = j), 0 (i \neq j)
δ
ij
⇒
1
(
i
=
j
)
,
0
(
i
=
j
)
•
A
=
(
a
b
c
d
)
A = \left( \begin{array}{rr} a & b \\ c & d \end{array} \right)
A
=
(
a
c
b
d
)
◦
⇒
A
2
−
(
a
+
d
)
A
+
(
a
d
−
b
c
)
I
=
0
\Rightarrow A^{2} - (a + d) A + (ad -bc) I = 0
⇒
A
2
−
(
a
+
d
)
A
+
(
a
d
−
b
c
)
I
=
0
행렬식
•
A
,
B
A, B
A
,
B
행렬,
r
,
s
∈
F
r, s \in \mathbb{F}
r
,
s
∈
F
◦
A
+
B
=
B
+
A
A + B = B + A
A
+
B
=
B
+
A
◦
A
+
(
B
+
C
)
=
(
A
+
B
)
+
C
A + (B + C) = (A + B) + C
A
+
(
B
+
C
)
=
(
A
+
B
)
+
C
◦
A
+
0
=
A
A + 0 = A
A
+
0
=
A
◦
A
+
(
−
A
)
=
0
A + (-A) = 0
A
+
(
−
A
)
=
0
◦
(
r
+
s
)
A
=
r
A
+
s
A
(r + s)A = rA + sA
(
r
+
s
)
A
=
r
A
+
s
A
◦
r
(
A
+
B
)
=
r
A
+
r
B
r(A + B) = rA + rB
r
(
A
+
B
)
=
r
A
+
r
B
◦
r
(
s
A
)
=
(
r
s
)
A
r(sA) = (rs)A
r
(
s
A
)
=
(
rs
)
A
◦
(
A
B
)
C
=
A
(
B
C
)
(AB)C = A(BC)
(
A
B
)
C
=
A
(
BC
)
◦
A
I
=
I
A
=
A
AI = IA = A
A
I
=
I
A
=
A
◦
(
A
+
B
)
C
=
A
C
+
B
C
(A + B)C = AC + BC
(
A
+
B
)
C
=
A
C
+
BC
◦
r
(
A
)
B
=
r
(
A
B
)
=
A
(
r
B
)
r(A)B = r(AB) = A(rB)
r
(
A
)
B
=
r
(
A
B
)
=
A
(
r
B
)
◦
A
n
:
=
A
⋅
A
n
−
1
A^{n} := A \cdot A^{n-1}
A
n
:=
A
⋅
A
n
−
1
◦
A
0
=
I
A^{0} = I
A
0
=
I
