线性代数与解析几何--第五章 线性变换.pdf

1ÊÙ ‚5C†
§ ‚5C†Vg
§ ‚5C†½Â
½Â. U,VêFþü‡‚5˜m,NA : U → V¡‚5N,X
Jé?¿x,y ∈ U, λ∈ F, Ñk:
A (x + y) = A (x) + A (y) ()
A (λx) = λA (x) ()
K¡A l‚5˜mU‚5˜mV‚5N. AO/,XJU = V,K¡A 
‚5˜mVþ˜‡‚5C†.
§ ‚5C†~f
~. rz‡•þNgC†E : E (x) = x, x ∈ V;±9rz‡•þN
"•þC†O: O(x) = θ, x ∈ V.
ØJwÑùü‡C†‚5C†. E ¡ü C†½ðC†, O¡"
C†.
n n
~. F êF¥nê|˜m, A = (ai j)n×nn
.C†A : F →
n 0 n n
F ½Â:éz‡x = (x1,x2,··· ,xn) ∈ F (ùp·‚^•þL«F ¥•þ),

a a ··· a x1
11 12 1n 

a21 a22 ··· a2nx2
A (x) = 
. 
··· ··· ··· ···. 
an1 an2 ··· ann xn
dÝ
¦{{KC†A ‚5C†.
N½C†´˜‡AÛVg,·‚F~)¹¥²~-º¡‡!^=
Ñ´‚5C†.
2 2 2
~. ~¥,·‚^•þL«R •Ã þ!.C†A : R Ã→ R!òz‡•þαN
x x
α'uX¶é¡•þ(„ã1). α= , KA (α) = .^Ý
L«Ò
y −y
´ Ã ! Ã !Ã !
³ x ´ 1 0 x
A (α) = A =
y 0 −1 y
d~ A ‚5C†,¡'uX¶º¡‡.
1
2 1ÊÙ ‚5C†
à !
x
C†A : R2 → R2´òz‡•þα_ž^=θC†.α= , A (α) =
y
à !
x˜
. ·‚|^Eê5OŽx˜,y˜. •þαéAEêr eiϕ. KA (α)éAE
y˜
êr ei(ϕ+θ).Ïd
x˜= r cos(ϕ+ θ) = r cosϕ cosθ− r sinϕ sinθ
= x cosθ− y sinθ
y˜= r sin(ϕ+ θ) = r cosϕ sinθ+ r sinϕ cosθ
= x sinθ+ y cosθ
^Ý
L«Ò´
à ! à !à !
³ x ´ cosθ−sinθ x
A (α) = A = .
y sinθ cosθ y
d~ A ‚5C†,¡^=C†.
e¡0·‚ÙG˜m¥˜
‚5C†.
~. 3Pn[x]¥,A ‡©Žf
d
A (p(x)) = p(x).
dx
d‡©{A ‚5C†.
~. 3d2
¤‚5˜m¥,齐
à !
αβ
M = ,
βδ
½ÂC† à ! à !à !
a b αβ a b
A =
c d γδ c d
dÝ
¦{{KØJwÑA ‚5C†.
§ ‚5C†5Ÿ
e¡·K‰Ñ
‚5C†˜
{ü5Ÿ.
·. A : V → V‚5C†,K
(1) A (θ) = θ; A (−a) = −A (a), a ∈ V
(2) eα1,α2,··· ,αmV¥‚5ƒ'•þ,KA (α1),A (α2),··· ,A (αm)‚5
ƒ'.
§ ‚5C†Ý
3
y²: (1) A (θ) = A (0 · a) = 0 · A (a) = θ;
A (−a) = A ((−1)a) = (−1)A (a) = −A (a).
(2) α1,α2,··· ,αm‚5ƒ',K3؏"λ1,λ2,··· ,λm ∈ F,¦
λ1α1 + λ2α2 + ··· + λmαm = θ.
ü>^‚5C†A Š^
,
λ1A (α1)+λ2A (α2)+···+λmA (αm) = A (λ1α1 +λ2α2 +···+λmαm) = A (θ) = θ
ÏdA (α1),A (α2),··· ,A (αm)‚5ƒ'.
2
ù‡·K`²‚5ƒ'•þ|