Solution Found!
Let be fixed numbers. The matrix below, called a
Chapter , Problem 11E(choose chapter or problem)
Problem 11E
Let be fixed numbers. The matrix below, called a Vandermonde matrix, occurs in applications such as signal processing, error-correcting codes, and polynomial interpolation.
Given
and define the polynomial
a. Show that .
We call an interpolating polynomial for the points because the graph of passes through the points.
b. Suppose are distinct numbers. Show that the columns of V are linearly independent. [Hint: How many zeros can a polynomial of degree n – 1 have?]
c. Prove: “If are distinct numbers, and are arbitrary numbers, then there is an interpolating polynomial of degree
Questions & Answers
QUESTION:
Problem 11E
Let be fixed numbers. The matrix below, called a Vandermonde matrix, occurs in applications such as signal processing, error-correcting codes, and polynomial interpolation.
Given
and define the polynomial
a. Show that .
We call an interpolating polynomial for the points because the graph of passes through the points.
b. Suppose are distinct numbers. Show that the columns of V are linearly independent. [Hint: How many zeros can a polynomial of degree n – 1 have?]
c. Prove: “If are distinct numbers, and are arbitrary numbers, then there is an interpolating polynomial of degree
ANSWER:
Solution 11E
a. As , using the values gives
Multiply the matrices on the left side and equate with the right hand side matrix gives
As . Compare with the above system of equations gives
.
