8.19 EXERCISES
whereUandVare given by (8.139) and (8.140) respectively andSis obtained by taking
the transpose ofSin (8.138) and replacing all the non-zero singular valuessiby 1/si. Thus,
Sreads
S=
1
4 00
0 13 0(^0012)
000
.
Substituting the appropriate matrices into the expression forxwe find
x=^18 (1111)T. (8.144)It is straightforward to show that this solves the set of equationsAx=bexactly, and
so the vectorb=(100)Tmust lie in the range ofA. This is, in fact, immediately
clear, sinceb=u^1. The solution (8.144) isnot, however, unique. There are three non-zero
singular values, butN= 4. Thus, the matrixAhas a one-dimensional null space, which
is ‘spanned’ byv^4 , the fourth column ofV, given in (8.140). The solutions to our set of
equations, consisting of the sum of the exact solution andanyvector in the null space of
A, therefore lie along the line
x=^18 (1111)T+α(1 − 11 −1)T,where the parameterαcan take any real value. We note that (8.144) is the point on this
line that is closest to the origin.
8.19 Exercises8.1 Which of the following statements about linear vector spaces are true? Where a
statement is false, give a counter-example to demonstrate this.
(a) Non-singularN×Nmatrices form a vector space of dimensionN^2.
(b) SingularN×Nmatrices form a vector space of dimensionN^2.
(c) Complex numbers form a vector space of dimension 2.
(d) Polynomial functions ofxform an infinite-dimensional vector space.
(e) Series{a 0 ,a 1 ,a 2 ,...,aN}for which∑N
n=0|an|(^2) =1formanN-dimensional
vector space.
(f) Absolutely convergent series form an infinite-dimensional vector space.
(g) Convergent series with terms of alternating sign form an infinite-dimensional
vector space.
8.2 Evaluate the determinants
(a)
∣
∣∣
∣
∣∣
ahg
hbf
gfc∣
∣∣
∣
∣∣, (b)∣∣
∣
∣∣
∣
∣
1023
01 − 21
3 − 34 − 2
− 21 − 21
∣∣
∣
∣∣
∣
∣
and(c)∣
∣
∣∣
∣
∣∣
gc ge a+ge gb+ge
0 bb b
ce e b+e
abb+fb+d