PROBLEM SET 9
Problem 1
Let xi and yj , i, j = 1, 2, . . . , n, be two sequences of numbers such that xi 6= yj
for every i and j. The entries of an n × n matrix C = (cij ) are defined by the
formula cij = 1/(xi − yj ) (such a matrix is called a Cauchy matrix). Prove that
Q
det C =
1≤i<j≤n (xj − xi )(yi −
Qn Qn
k=1
l=1 (xk − yl )
yj )
.
Problem 2
Let cj , j = 1, 2, . . . , n, be positive numbers such that ci 6= cj when i 6= j. Prove
that the functions fj (x) = sin(cj x), j = 1, 2, . . . , n, are linearly independent in the
space of continuous functions on the interval [0, 1].
From the book: sec. 4.3: 23, 27, 28, sec 4.4: 6; sec 5.1: 2(c,e), 3(a,b,c).
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