数值分析1-MathBasics_169109569

x = a1 x1 + a2 x2 + an xn ,
5L), u{x } ¨¤m'|. dm'n. 5 m'S7, 'tvk9X.
(
n i i=1
Example
PN . {1, x, , x N } Example
¨¤|. dim P
N
= N + 1.
C [a, b]. N, {1, x, , x N }
x
u(S, || ||)Dm. TêSp'ê.
xS< (ê) ê:
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||x|| =
(x, x).
S~
(R n , (, )).
S
n
(x, y ) =
i=1
xi yi .
xS< (ê)
ê:
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S~
(R n , (, )).
S
n
(x, y ) =
xi yi .
i=1
(C [a, b], R).
C [a, b] := {f |f is continuous in [a, b]}
xS< (ê)
ê:
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5m~f
(R n , R), (C n , C ), (C n , R), (R n , C )?
[a, b]4m. uC [a, b]5m. êm:
C [a, b] := {f |f is continuous in [a, b]}
i i i
a1 x1 + a2 x2 + + an xn = 0,
u{x } 59. u, 59.
n i i=1
xS< (ê)
ê:
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5'
(S, P)5m. x ∈ S, i = 1, 2, , n. Xt Q|êa ∈ P, i = 1, 2, , n, |a | > 0, ÷v
4 Matrix Spaces
xS< (ê)
ê:
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5m
S 8, P ê(R C ). üN9X + : S × S → S : P × S → S , ÷v
(a + b) + c = a + (b + c) a+0=a a + (a) = 0 a+b =b+a (λ + ) a = λ a + a λ (a + b) = λ a + λ b (λ) a = λ ( a)
PN := {p(x)|p(x) = a0 + a1 x + + aN x N , ai ∈ R(C )}
xS< (ê)
ê:
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5m~f
(R n , R), (C n , C ), (C n , R), (R n , C )?
[a, b]4m. uC [a, b]5m. êm:
C [a, b] := {f |f is continuous in [a, b]}
u(S, P)¨¤5m.
xS< (ê) ê:
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1a=a
5m~f
(R n , R), (C n , C ), (C n , R), (R n , C )?
xS< (ê)
ê:
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5m~f
(R n , R), (C n , C ), (C n , R), (R n , C )?
[a, b]4m. uC [a, b]5m.
i i i
a1 x1 + a2 x2 + + an xn = 0,
u{x } 59. u, 59.
n i i=1
Example PN
{1, x, , x }59'.
N
xS< (ê)
ê:
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5'
(S, P)5m. x ∈ S, i = 1, 2, , n. Xt Q|êa ∈ P, i = 1, 2, , n, |a | > 0, ÷v
a
xS< (ê)
ê:
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Sm5'§{
S Sm. x , x , , x
÷
1 2 n
∈S
59', ¨a¨ e'
(x1 , xn ) (x2 , xn ) .. . (xn , xn ) ,
Gn =
(x1 , x1 ) (x1 , x2 ) (x2 , x1 ) (x2 , x2 ) .. .. . . (xn , x1 ) (xn , x2 )
ê:
xS<
êX u
Email: czheng@
j
1 Linear Space
2 Norm
3 Inner Product
4 Matrix Spaces
xS< (ê)
ê:
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Outline
1 Linear Space
2 Norm
3 Inner Product
y ', e(x, y ) = 0.
xS< (ê)
ê:
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S
(S, P)5m. SN S × S → P, ÷v
(x, x) ≥ 0. (x, x) = 0 x = 0. (x, y ) = (y , x) (ax + by , z) = a(x, z) + b(y , z)
y ', e(x, y ) = 0. pê
i i i
a1 x1 + a2 x2 + + an xn = 0,
u{x } 59. u, 59.
n i i=1
Example PN
{1, x, , x }59'.
N
Example
Cper [π, π] {1, cos x, sin x, , cos nx, sin nx}
59.
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`, det G
n
= 0.
xS< (ê)
ê:
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IO
{x , x , , x }Sm S '|. ué?x ∈ S, Q' |êa , a , , a ÷v
1 2 n 1 2 n
x = a1 x1 + a2 x2 + + an xn .
xS< (ê)
ê:
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IO
{x , x , , x }Sm S '|. ué?x ∈ S, Q' |êa , a , , a ÷v
1 2 n 1 2 n
x = a1 x1 + a2 x2 + + an xn .
K XO a , a , , a ?
1 2 n
xS< (ê)
ê:
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IO
ü>Ox S
∞ 1 2 n
nê: ∞-, 1-, 2-ê
5XQx ∈ S, ÷v
Example R
||xn xn+p || < , p ∈ N, n > N.
||xn x|| → 0, n → +∞.
'.
xS< (ê) ê:
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C [a, b]
?ü1d. S, (C [a, b], || || ) ', (C [a, b], || || ) (C [a, b], || || ). 5? (S, || ||)D5m. {x } CauchyS, i.e., > 0, N ∈ N, ÷v
PN := {p(x)|p(x) = a0 + a1 x + + aN x N , ai ∈ R(C )} N-
0
m: R
N×N
xS< (ê)
ê:
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5'
(S, P)5m. x ∈ S, i = 1, 2, , n. Xt Q|êa ∈ P, i = 1, 2, , n, |a | > 0, ÷v
59'. dim C [a, b] = +∞.
ê:
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xS< (ê)
Outline
1 Linear Space
2 Norm
3 Inner Product
4 Matrix Spaces
xS< (ê)
ê:
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êDm
|| || : S → R N, ÷v
||f || ≥ 0, f ∈ S. ||f || = 0 f = 0
xS< (ê)
ê:
{x } 59, éx ∈ S, k{a } , &
n i i=1 n i i=1
x = a1 x1 + a2 x2 + an xn , (
5L), u{x } ¨¤m'|. dm'n.
n i i=1
xS< (ê)
ê:
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{x } 59, éx ∈ S, k{a } , &
2 Norm
3 Inner Product
4 Matrix Spaces
xS< (ê)
ê:
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S
(S, P)5m. SN S × S → P, ÷v
(x, x) ≥ 0. (x, x) = 0 x = 0. (x, y ) = (y , x) (ax + by , z) = a(x, z) + b(y , z) x