Page 197 - Matrix Analysis & Applied Linear Algebra
P. 197
192 Chapter 4 Vector Spaces
4.3.7. Determine whether or not the following set of matrices is a linearly
independent set:
10 11 11 11
, , , .
00 00 10 11
4.3.8. Without doing any computation, determine whether the following ma-
trix is singular or nonsingular:
n 1 1 ··· 1
1 n 1 ··· 1
A = 1 1 n ··· 1 .
. . . . . . . . .
. . . . . .
1 1 1 ··· n
n×n
4.3.9. In theory, determining whether or not a given set is linearly independent
is a well-defined problem with a straightforward solution. In practice,
however, this problem is often not so well defined because it becomes
clouded by the fact that we usually cannot use exact arithmetic, and con-
tradictory conclusions may be produced depending upon the precision
of the arithmetic. For example, let
.1 .2 .3
,
,
S = .4 .5 .6 .
.7 .8 .901
(a) Use exact arithmetic to determine whether or not S is linearly
independent.
(b) Use 3-digit arithmetic (without pivoting or scaling) to determine
whether or not S is linearly independent.
n
4.3.10. If A m×n is a matrix such that a ij = 0 for each i =1, 2,...,m
j=1
(i.e., each row sum is 0), explain why the columns of A are a linearly
dependent set, and hence rank (A) <n.
m×1
4.3.11. If S = {u 1 , u 2 ,..., u n } is a linearly independent subset of , and
if P m×m is a nonsingular matrix, explain why the set
P(S)= {Pu 1 , Pu 2 ,..., Pu n }
must also be a linearly independent set. Is this result still true if P is
singular?
