线性代数与解析几何--第六章 欧氏空间.pdf

文档介绍：
6
3

R ,
n×m

P [ ]
×
P

n n x n m
(
)


3

R
3
R

–

Æ
§
3

R
a
|a|
|b|
(
b,
θ

)
(a, b)=|a||b| cos θ.
Æ

a
a
b

|a| = (a, a)
(a, b)
θ= os 
(a, a)(b, b)

Æ

1.


R
a
b

V V
(a b),

,

a b ∈,
(1)
, V
(a, b)=(b, a)
1
a b c ∈,
(2)
λ, , V
(λa, b)=λ(a, b)(a + b, c)=(a, c)+(b, c)

a ∈,
(a a) ≥ 0,
a = 0(
(3)
V ,

).

(a b)
a
b
,
R

, V

(Euclid)
,
.

(1)
(2)
a b c ∈,

λ1,λ2 , , V
(λ1a + λ2b, c)=λ1(a, c)+λ2(b, c)
(a,λ1b + λ2c)=λ1(a, b)+λ2(a, c)

k m k m
( λiai, μjbj)= λiμj(ai, bj)
i=1 j=1 i=1 j=1

(1)
(2)
(a, 0)=0
0

(2)
(a b)
(1)
,

(a b)


V V , V

(3)
(a a)
,

(a b),
(a a)

V f , f ,


V
(a, b)=f(a, b)

Æ
R
V

2.


V V

2
(
–
):
(· ·)

V , V V

a
b,


|(a, b)|≤(a, a)(b, b)

a
b
,

λ
0 ≤(λa + b,λa + b)
=