Page 190 - Matrix Analysis & Applied Linear Algebra

4.3 Linear Independence 185
Example 4.3.4

Vandermonde Matrices. Matrices of the form

1 x 1 x 1 ··· x 1
 2 n−1 
 1 x 2 x 2 2 ··· x n−1 
2
V m×n =  . . . . 
 . . . .
. . . ··· . 
1 x 2 ··· x n−1
x m
m m
26
in which x i = x j for all i = j are called Vandermonde matrices.
Problem: Explain why the columns in V constitute a linearly independent set
whenever n ≤ m.
Solution: According to (4.3.2), the columns of V form a linearly independent
set if and only if N (V)= {0}. If

1 x ··· x 0
 2 n−1     
x 1
1 1 α 0
 1 x 2 x 2 2 ··· x n−1   α 1   0 
2
.
 . . . .   .  =   , (4.3.8)
 . . . .   .   . 
. . . ··· . . .
1 x m x 2 m ··· x n−1 α n−1 0
m
then for each i =1, 2,...,m,
2
α 0 + x i α 1 + x α 2 + ··· + x n−1 α n−1 =0.
i
i
This implies that the polynomial
2
p(x)= α 0 + α 1 x + α 2 x + ··· + α n−1 x n−1
has m distinct roots—namely, the x i ’s. However, deg p(x) ≤ n − 1 and the
fundamental theorem of algebra guarantees that if p(x) is not the zero polyno-
mial, then p(x) can have at most n − 1 distinct roots. Therefore, (4.3.8) holds
if and only if α i = 0 for all i, and thus (4.3.2) insures that the columns of V
form a linearly independent set.

26
This is named in honor of the French mathematician Alexandre-Theophile Vandermonde (1735–
1796). He made a variety of contributions to mathematics, but he is best known perhaps for
being the first European to give a logically complete exposition of the theory of determinants.
He is regarded by many as being the founder of that theory. However, the matrix V (and
an associated determinant) named after him, by Lebesgue, does not appear in Vandermonde’s
published work. Vandermonde’s first love was music, and he took up mathematics only after
he was 35 years old. He advocated the theory that all art and music rested upon a general
principle that could be expressed mathematically, and he claimed that almost anyone could
become a composer with the aid of mathematics.