Page 195 - Matrix Analysis & Applied Linear Algebra

190 Chapter 4 Vector Spaces

2 n
For example, to verify that the set of polynomials P = 1,x,x ,...,x is
linearly independent, observe that the associated Wronski matrix

1 x x ··· x
 2 n 
 0 1 2x ··· nx n−1 

W(x)=  0 0 2 ··· n(n − 1)x n−2 

 . . . . 
 . . . . . . 
. . . . .
0 0 0 ··· n!
is triangular with nonzero diagonal entries. Consequently, W(x) is nonsingular
for every value of x, and hence P must be an independent set.

Exercises for section 4.3

4.3.1. Determine which of the following sets are linearly independent. For those
sets that are linearly dependent, write one of the vectors as a linear
combination of the others.
     
1 2 1
 
1
(a)       ,
,
5
,
2
3 0 9
 
(b) {(123 ) , (045 ) , (006 ) , (111 )} ,
     
3 1 2
 
(c)       ,
1
2
,
,
0
1 0 0
 
(d) {(2222 ) , (2202 ) , (2022 )} ,
1 0 0 0
       
 
 
 
 2   2   2   2 
 
       
 
 0   0   1   0 
(e)         .
 4  ,  4  ,  4  ,  4 
       
 
 0   1   0   0 
 
       
 3 3 3 3 
 
 
0 0 0 1
2 1 1 0

4.3.2. Consider the matrix A = 4 2 1 2 .
6 3 2 2
(a) Determine a maximal linearly independent subset of columns
from A.
(b) Determine the total number of linearly independent subsets that
can be constructed using the columns of A.