szTom/LADRSolutions
1
0
Fork 0
mirror of https://github.com/szdytom/LADRSolutions.git synced 2025-10-19 16:30:16 +00:00
Code Issues Packages Projects Releases Wiki Activity

2A p2

Browse Source 
Signed-off-by: szdytom <szdytom@qq.com>
This commit is contained in:
方而静 2025-07-27 16:58:57 +08:00
parent 7bdedf4c6f
commit 0450974138
Signed by: szTom
GPG Key ID: 072D999D60C6473C
2 changed files with 34 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

33
sections/3A.typ
View File

@ -1,4 +1,5 @@
#import "../styles.typ": exercise_sol, note, tab
#import "../math.typ": Poly
#exercise_sol(type: "proof")[
设 $b, c in RR$。定义
@ -38,3 +39,35 @@
#tab 综上所述，$T$ 是线性映射，当且仅当 $b = c = 0$。
]
#exercise_sol(type: "proof")[
设 $b, c in RR$。定义
$ T:& Poly(RR) -> RR^2 \ &p |-> vec(3p(4) + 5p'(6) + b p(1)p(2), integral_(-1)^2 x^3 p(x) dif x + c sin p(0)) $
证明：$T$ 是线性映射，当且仅当，$b = c = 0$。
][
首先，假设 $b = c = 0$。则 $T(p) = (3p(4) + 5p'(6), integral_(-1)^2 x^3 p(x) dif x)$，我们逐条验证线性映射的定义（原书3.1）中的要求：
/ 可加性: 对任意 $p, q in Poly(RR)$，$T(p + q) = T p + T q$。 \
证明：设 $p, q in Poly(RR)$，则
$ T(p + q) &= (3(p + q)(4) + 5(p + q)'(6), integral_(-1)^2 x^3 (p + q)(x) dif x) \
&= (3p(4) + 3q(4) + 5p'(6) + 5q'(6), integral_(-1)^2 x^3 p(x) dif x + integral_(-1)^2 x^3 q(x) dif x) \
&= (3p(4) + 5p'(6), integral_(-1)^2 x^3 p(x) dif x) + (3q(4) + 5q'(6), integral_(-1)^2 x^3 q(x) dif x) \
&= T p + T q $
/ 齐次性: 对任意 $p in Poly(RR)$ 和任意 $lambda in RR$，$T(lambda p) = lambda T p$。 \
证明：设 $p in Poly(RR)$，则
$ T(lambda p) &= (3(lambda p)(4) + 5(lambda p)'(6), integral_(-1)^2 x^3 (lambda p)(x) dif x) \
&= (lambda (3p(4)) + lambda (5p'(6)), lambda integral_(-1)^2 x^3 p(x) dif x) \
&= lambda (3p(4), integral_(-1)^2 x^3 p(x) dif x) \
&= lambda T p $
#tab 综上，$T$ 满足线性映射的定义。
#tab 另一方面，假设 $T$ 是线性映射。设 $p: x |-> x + 1$。则根据齐次性的要求，有
$ T(2p) = (40 + 24b, 207 / 10 + c sin 2) = (40 + 12b, 207 / 10 + 2c sin 1) = 2 T(p) $
#tab 解得 $b = c = 0$。这就是说 $T$ 是线性映射，当且仅当 $b = c = 0$。
]

1
styles.typ
View File

@ -99,6 +99,7 @@
))
set footnote(numbering: "注1")
show math.equation: set text(font: mathfont)
set math.vec(delim: ("[", "]"))
hide[#title <book-title>]