向量通常用頭上有一個箭頭的小寫字母表示，如
x
⃗
\vec{x}
x
x
⃗
=
(
8
6
7
5
3
)
\vec{x} = \left(\begin{array}{cc} 8\\ 6\\ 7\\ 5\\ 3 \end{array} \right)
x
= ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ 8 6 7 5 3 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
x
1
=
8
,
x
2
=
6
x_1=8,x_2=6
x 1 = 8 , x 2 = 6
x
⃗
=
[
8
,
6
,
7
,
5
,
3
]
\vec{x}=[8, 6, 7, 5, 3]
x
= [ 8 , 6 , 7 , 5 , 3 ]
n
n
n
x
⃗
\vec{x}
x
n
n
n
x
i
x_i
x i
i
i
i
x
⃗
\vec{x}
x
矩陣和表格很像。矩陣由行（row，橫向的列表）和列（column，縱向的列表）組成。
A
=
[
17
18
5
5
45
1
42
28
30
15
115
3
10
10
10
21
51
2
28
5
65
39
132
5
24
26
45
21
116
4
]
A= \left[\begin{array}{cc} 17 & 18 & 5 & 5 & 45 & 1\\ 42 & 28 & 30 & 15 & 115 & 3\\ 10 & 10 & 10 & 21 & 51 & 2\\ 28 & 5 & 65 & 39 & 132 & 5\\ 24 & 26 & 45 & 21 & 116 & 4 \end{array} \right]
A = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 1 7 4 2 1 0 2 8 2 4 1 8 2 8 1 0 5 2 6 5 3 0 1 0 6 5 4 5 5 1 5 2 1 3 9 2 1 4 5 1 1 5 5 1 1 3 2 1 1 6 1 3 2 5 4 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
5
×
6
5 \times 6
5 × 6
A
A
A
m
m
m
n
n
n
a
a
a
a
a
a
i
i
i
j
j
j
a
45
=
132
a_{45}=132
a 4 5 = 1 3 2
i
i
i
j
j
j
a
i
j
a_{ij}
a i j
i
j
−
e
n
t
r
y
ij-entry
i j − e n t r y
i
j
−
c
o
m
p
n
e
n
t
ij-compnent
i j − c o m p n e n t
m
,
n
m,n
m , n
a
i
j
,
i
=
1
,
.
.
.
,
m
,
j
=
1
,
.
.
.
,
n
a_{ij},i=1,...,m,j=1,...,n
a i j , i = 1 , . . . , m , j = 1 , . . . , n
A
=
[
a
11
.
.
.
a
1
j
.
.
.
a
1
n
.
.
.
a
i
1
.
.
.
a
i
j
.
.
.
a
i
n
.
.
.
a
m
1
.
.
.
a
m
j
.
.
.
a
m
n
]
A= \left[\begin{array}{cc} a_{11} & ... & a_{1j} & ... & a_{1n}\\ . & & . & & .\\ a_{i1} & ... & a_{ij} & ... & a_{in}\\ . & & . & & .\\ a_{m1} & ... & a_{mj} & ... & a_{mn} \end{array} \right]
A = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ a 1 1 . a i 1 . a m 1 . . . . . . . . . a 1 j . a i j . a m j . . . . . . . . . a 1 n . a i n . a m n ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
m
×
n
m\times n
m × n
a
i
j
a_{i}j
a i j
A
(
i
)
=
(
a
i
1
,
.
.
.
,
a
(
i
n
)
)
A_{(i)} = (a_{i1}, ..., a_{(in)})
A ( i ) = ( a i 1 , . . . , a ( i n ) )
i
i
i
A
(
j
)
=
(
a
1
j
,
.
.
.
,
a
(
m
j
)
)
A^{(j)} = (a_{1j}, ..., a_{(mj)})
A ( j ) = ( a 1 j , . . . , a ( m j ) )
j
j
j
n
n
n
1
×
n
1\times n
1 × n
向量
v
⃗
\vec{v}
v
∣
v
⃗
∣
|\vec{v}|
∣ v
∣
∣
v
⃗
∣
=
∑
i
=
1
n
x
i
2
|\vec{v}| = \sqrt{\sum_{i=1}^{n}x_i^2}
∣ v
∣ = i = 1 ∑ n x i 2
v
⃗
=
[
4
,
11
,
8
,
10
]
\vec{v}=[4,11,8,10]
v
= [ 4 , 1 1 , 8 , 1 0 ]
∣
v
⃗
∣
=
4
2
+
1
1
2
+
8
2
+
1
0
2
=
301
|\vec{v}|=\sqrt{4^2+11^2+8^2+10^2}=\sqrt{301}
∣ v
∣ = 4 2 + 1 1 2 + 8 2 + 1 0 2
= 3 0 1
將兩個向量相加，就是把
v
1
⃗
\vec{v_1}
v 1
v
2
⃗
\vec{v_2}
v 2
A
=
[
a
1
,
a
2
,
.
.
.
,
a
n
]
,
B
=
[
b
1
,
b
2
,
.
.
.
,
b
n
]
A = [a_1, a_2, ..., a_n], B=[b_1,b_2,...,b_n]
A = [ a 1 , a 2 , . . . , a n ] , B = [ b 1 , b 2 , . . . , b n ]
A
+
B
=
[
a
1
+
b
1
,
a
2
+
b
2
,
.
.
.
,
a
n
+
b
n
]
A+B=[a_1+b_1, a_2+b_2, ..., a_n+b_n]
A + B = [ a 1 + b 1 , a 2 + b 2 , . . . , a n + b n ]
將一個標量（實數）和一個向量相乘，就是把向量裏面的每個元素與這個標量相乘。如果
d
d
d
v
⃗
\vec{v}
v
v
1
,
v
2
,
.
.
.
,
v
n
v_1, v_2, ...,v_n
v 1 , v 2 , . . . , v n
d
∗
v
⃗
=
[
d
v
1
,
d
v
2
,
.
.
.
,
d
v
n
]
d*\vec{v}=[dv_1,dv_2,...,dv_n]
d ∗ v
= [ d v 1 , d v 2 , . . . , d v n ]
兩個向量的內積代表將兩個向量的每個對應的元素相乘然後將這些結果相加，表示爲
(
v
1
⃗
,
v
2
⃗
)
(\vec{v_1}, \vec{v_2})
( v 1
, v 2
)
v
1
⃗
⋅
v
2
⃗
\vec{v_1}\cdot\vec{v_2}
v 1
⋅ v 2
(
v
1
⃗
,
v
2
⃗
)
=
v
1
⃗
⋅
v
2
⃗
=
∑
i
=
1
n
x
i
y
i
(\vec{v_1}, \vec{v_2})=\vec{v_1}\cdot\vec{v_2}=\sum_{i=1}^{n}x_i y_i
( v 1
, v 2
) = v 1
⋅ v 2
= ∑ i = 1 n x i y i
如果兩個向量的內積爲0，那麼這兩個向量是正交的。
向量長度爲1的向量可以叫做單位向量。任何向量長度大於0的向量可以通過將每個元素除以向量長度的方法正規化（normalized）爲單位向量。
v
⃗
=
[
2
,
4
,
1
,
2
]
\vec{v} = [2, 4, 1, 2]
v
= [ 2 , 4 , 1 , 2 ]
∣
v
⃗
∣
=
2
2
+
4
2
+
1
2
+
2
2
=
5
|\vec{v}|=\sqrt{2^2 + 4^2 + 1^2 + 2^2} = 5
∣ v
∣ = 2 2 + 4 2 + 1 2 + 2 2
= 5
u
⃗
=
[
2
/
5
,
4
/
5
,
1
/
5
,
1
/
5
]
\vec{u}=[2/5, 4/5, 1/5, 1/5]
u
= [ 2 / 5 , 4 / 5 , 1 / 5 , 1 / 5 ]
∣
u
⃗
∣
=
(
2
/
5
)
2
+
(
4
/
5
)
2
+
(
1
/
5
)
2
+
(
1
/
5
)
2
=
1
|\vec{u}| = \sqrt{(2/5)^2 + (4/5)^2 + (1/5)^2 + (1/5)^2}=1
∣ u
∣ = ( 2 / 5 ) 2 + ( 4 / 5 ) 2 + ( 1 / 5 ) 2 + ( 1 / 5 ) 2
= 1
兩個正交的單位向量叫做orthonormal。
將一組向量變成互相正交的單位向量
A
=
[
1
2
1
0
2
0
2
3
1
1
1
0
]
A= \left[\begin{array}{cc} 1 & 2 & 1\\ 0 & 2 & 0\\ 2 & 3 & 1\\ 1 & 1 & 0 \end{array}\right]
A = ⎣ ⎢ ⎢ ⎡ 1 0 2 1 2 2 3 1 1 0 1 0 ⎦ ⎥ ⎥ ⎤
A
=
[
6
6
2
6
2
3
0
2
2
3
−
1
3
6
3
0
0
6
6
−
2
6
−
2
3
]
A= \left[\begin{array}{cc} \frac{\sqrt{6}}{6} & \frac{\sqrt{2}}{6} & \frac{2}{3}\\ 0 & \frac{2\sqrt{2}}{3} & -\frac{1}{3}\\ \frac{\sqrt{6}}{3} & 0 & 0\\ \frac{\sqrt{6}}{6} & -\frac{\sqrt{2}}{6} & -\frac{2}{3} \end{array}\right]
A = ⎣ ⎢ ⎢ ⎢ ⎡ 6 6
0 3 6
6 6
6 2
3 2 2
0 − 6 2
3 2 − 3 1 0 − 3 2 ⎦ ⎥ ⎥ ⎥ ⎤
v
1
⃗
=
[
1
,
0
,
2
,
1
]
\vec{v_1}=[1,0,2,1]
v 1
= [ 1 , 0 , 2 , 1 ]
u
1
⃗
=
[
1
6
,
0
,
2
6
,
1
6
]
\vec{u_1}=[\frac{1}{\sqrt{6}}, 0, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}]
u 1
= [ 6
1 , 0 , 6
2 , 6
1 ]
w
2
⃗
=
v
2
⃗
−
u
1
⃗
⋅
v
2
⃗
∗
u
1
⃗
=
[
2
,
2
,
3
,
1
]
−
[
1
6
,
0
,
2
6
,
1
6
]
⋅
[
2
,
2
,
3
,
1
]
∗
[
1
6
,
0
,
2
6
,
1
6
]
=
[
2
,
2
,
3
,
1
]
−
9
6
∗
[
1
6
,
0
,
2
6
,
1
6
]
=
[
2
,
2
,
3
,
1
]
−
[
3
2
,
0
,
3
,
3
2
]
=
[
1
2
,
2
,
0
,
−
1
2
]
\begin{aligned} \vec{w_2}=\vec{v_2}-\vec{u_1}\cdot\vec{v_2}*\vec{u_1}=&[2, 2, 3, 1] - [\frac{1}{\sqrt{6}}, 0, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}]\cdot[2,2,3,1]*[\frac{1}{\sqrt{6}}, 0, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}]\\ =&[2,2,3,1]-\frac{9}{\sqrt{6}}*[\frac{1}{\sqrt{6}}, 0, \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}]\\ =&[2,2,3,1]-[\frac{3}{2},0,3,\frac{3}{2}]\\ =&[\frac{1}{2}, 2, 0, -\frac{1}{2}] \end{aligned}
w 2
= v 2
− u 1
⋅ v 2
∗ u 1
= = = = [ 2 , 2 , 3 , 1 ] − [ 6
1 , 0 , 6
2 , 6
1 ] ⋅ [ 2 , 2 , 3 , 1 ] ∗ [ 6
1 , 0 , 6
2 , 6
1 ] [ 2 , 2 , 3 , 1 ] − 6
9 ∗ [ 6
1 , 0 , 6
2 , 6
1 ] [ 2 , 2 , 3 , 1 ] − [ 2 3 , 0 , 3 , 2 3 ] [ 2 1 , 2 , 0 , − 2 1 ]
w
2
⃗
\vec{w_2}
w 2
v
2
⃗
=
[
2
6
,
2
2
3
,
0
,
−
2
6
]
\vec{v_2}=[\frac{\sqrt{2}}{6}, \frac{2\sqrt{2}}{3}, 0, -\frac{\sqrt{2}}{6}]
v 2
= [ 6 2
, 3 2 2
, 0 , − 6 2
]
u
3
⃗
=
[
2
3
,
−
1
3
,
0
,
−
2
3
]
\vec{u_3}=[\frac{2}{3}, -\frac{1}{3}, 0, -\frac{2}{3}]
u 3
= [ 3 2 , − 3 1 , 0 , − 3 2 ]
w
k
⃗
=
v
k
⃗
−
∑
i
=
1
k
−
1
u
t
⃗
⋅
v
k
⃗
∗
u
t
⃗
\vec{w_k}=\vec{v_k}-\sum_{i=1}^{k-1}\vec{u_t}\cdot\vec{v_k}*\vec{u_t}
w k
= v k
− i = 1 ∑ k − 1 u t
⋅ v k
∗ u t
如果一個矩陣的行數和列數相同，那麼這個矩陣是方塊矩陣。
一個矩陣的轉置矩陣是通過將原本的矩陣的每個元素交換行和列來生成的。
A
A
A
A
T
A^T
A T
A
=
[
1
2
3
4
5
6
]
A= \left[\b-12.667 15.667-32.333 47-59 H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359 c-16-25.333-24-45-24-59z">
⋅ v k
∗ u t
如果一個矩陣的行數和列數相同，那麼這個矩陣是方塊矩陣。
一個矩陣的轉置矩陣是通過將原本的矩陣的每個元素交換行和列來生成的。
A
A
A
A
T
A^T
A T
A
=
[
1
2
3
4
5
6
]
A= \left[\begin{array}{cc} 1 & 2 & 3\\ 4 & 5 & 6 \end{array}\right]
A = [ 1 4 2 5 3 6 ]
A
A
A
A
T
=
[
1
4
2
5
3
6
]
A^T= \left[\begin{array}{cc} 1 & 4\\ 2 & 5\\ 3 & 6 \end{array}\right]
A T