南航双语矩阵论matrixtheory第三章部分题解

Solution Key to Some Exercises in Chapter 3

#5. Determine the kernel and range of each of the following linear transformations on 2P

(a) (())'()p x xp x σ=

(b) (())()'()p x p x p x σ=- (c) (())(0)(1)p x p x p σ=+

Solution (a) Let ()p x ax b =+. (())p x ax σ=.

(())0p x σ= if and only if 0ax = if and only if 0a =. Thus, ker(){|}b b R σ=∈

The range of σis 2()P σ={|}ax a R ∈ (b) Let ()p x ax b =+. (())p x ax b a σ=+-.

(())0p x σ= if and only if 0ax b a +-= if and only if 0a =and 0b =. Thus, ker(){0}σ=

The range of σis 2()P σ=2{|,}P ax b a a b R +-∈=

(c) Let ()p x ax b =+. (())p x bx a b σ=++.

(())0p x σ= if and only if 0bx a b ++= if and only if 0a =and 0b =. Thus, ker(){0}σ=

The range of σis 2()P σ=2{|,}P bx a b a b R ++∈= 备注: 映射的核以及映射的像都是集合,应该以集合的记号来表达或者用文字来叙述. #7. Let be the linear mapping that maps 2P into 2R defined by

10

()(())(0)p x dx p x p σ⎛⎫

⎪= ⎪⎝⎭

⎰ Find a matrix A such that

()x A ασαββ⎛⎫

+= ⎪⎝⎭

.

Solution

1(1)1σ⎛⎫

= ⎪⎝⎭ 1/2()0x σ⎛⎫

= ⎪⎝⎭

11/211/2()1010x ασαβαββ⎛⎫⎛⎫

⎛⎫⎛⎫

+=+= ⎪ ⎪

⎪⎪⎝⎭⎝⎭⎝⎭⎝⎭

Hence, 11/21

0A ⎛⎫

= ⎪⎝⎭

#10. Let σ be the transformation on 3P defined by

(())'()"()p x xp x p x σ=+

a) Find the matrix A representing σ with respect to 2[1,,]x x b) Find the matrix B representing σ with respect to 2[1,,1]x x + c) Find the matrix S such that 1B S AS -=

d) If 2012()(1)p x a a x a x =+++, calculate (())n p x σ. Solution (a) (1)0σ= ()x x σ=

22()22x x σ=+

002010002A ⎛⎫

⎪

= ⎪ ⎪⎝⎭

(b) (1)0σ=

()x x σ=

22(1)2(1)x x σ+=+

000010002B ⎛⎫

⎪

= ⎪ ⎪⎝⎭

(c)

2[1,,1]x x +2[1,,]x x =101010001⎛⎫

⎪

⎪ ⎪⎝⎭

The transition matrix from 2[1,,]x x to 2[1,,1]x x + is

101010001S ⎛⎫ ⎪= ⎪ ⎪⎝⎭

, 1

B S AS -= (d) 2201212((1))2(1)n n a a x a x a x a x σ+++=++

#11. Let A and B be n n ⨯ matrices. Show that if A is similar to B then there exist n n ⨯

matrices S and T , with S nonsingular, such that A ST =and B TS =.

Proof There exists a nonsingular matrix P such that 1A P BP -=. Let 1S P -=, T BP =. Then A ST =and B TS =.

#12. Let σ be a linear transformation on the vector space V of dimension n . If there exist a vector v such that 1()v 0n σ-≠ and ()v 0n σ=, show that

(a) 1,(),,()v v v n σσ-L are linearly independent.

(b) there exists a basis E for V such that the matrix representing σ with respect to the basis E is

00001

0000

010⎛⎫

⎪

⎪

⎪ ⎪⎝⎭

L L M M M M L

Proof