南航双语矩阵论 matrix theory第二章部分题解

Solution Key to Some Exercises in Chapter 2 #3.

(a) It is a unit circle centered at the origin.

coordinate axes.

(b) It is a square of unit side length, centered at the origin and with vertices on the

(c) It is a square of unit side length, centered at the origin and with sides parallel to coordinate axes.

#12 The process will fail.

u1

v1v1

p1 v2,u1 u1

v2 p1 0

,

u2

v2 p1v2 p1

When constructing u3, we have to compute p2 v3,u1 u1 v3,u2 u2.

Since v3 Span{v1,v2} Span{u1,u2}, we obtain that v3 v3,u1 u1 v3,u2 u2. Thus v3 p2 0. The process fails.

#14 (b) 1

2x 1

form an orthonormal basis for S.

(c) 1

and

x 1)

The best least squares approximation

to subspace S, which is

1

x 1)

x 1)

is given by the projection

of

415

onto the

45

x

#16 (b) The solutions to the least squares problem

AbAAx

H

H

Ax b

H

satisfy .

. That is AHbx since

AA I

Replace b by each given vector to get the solution.

#18

U

is orthogonal if and only if

u1T Tu2

(u1,u2, ,un)

T u n

U

1

U

T

. Thus

UU

T

I

UU

T

TT uu u2u 112

n

un u

T

This proves the result.

#21 (a) It does not define a norm in C[a,b].

Let f(x) (x a)(x b) C[a,b]. Then f 0, but f is not the zero function in

C[a,b]. The positive definiteness is violated.

(b) and (c) define norms in C[a,b].

Remark: When check the conditions in the definition of norm, some students forgot to

verify the condition that f 0 if and only if f 0.

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