Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING...

Give a proof, base the proof on the Determinant of a Vandermonde matrix that the INTERPOLATING POLYNOMIAL exist and its unique.

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Expert Solution

Suppose that the interpolation matrix exists and is in the form :

Now, the statement that interpolates the data points mean that:

Substituting this in the first equation, we get:

We need to solve for the s to construct . Since such a construction is possible, by reversing the argument, we see that the interpolating polynomial exists. Also, the matrix on the left is called a Vandermonde matrix.

To prove it's uniqueness, we write the above matrix equation as:

We have that is non-singular, because:

because the points are distinct, the determinant can't be zero as is never zero, therefore is nonsingular and the system has a unique solution .

Colby Messinger answered 1 year ago

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