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which checks.
There is another approach to solving 1, based on the Fast Fourier Transform, that also requires an amount of work proportional to
. The point for now is to see that the use of linear transformations on can help us perform computations involving
polynomials. The original problem—to fit a polynomial of minimum degree to a set of data points—was not couched in the
language of linear algebra at all. But rephrasing it in those terms and using matrices and the notation of linear transformations on
has allowed us to see when a unique solution must exist, how to compute it efficiently, and how to transform it among
various forms.
Exercise Set 4.4
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Identify the operations on polynomials that correspond to the following operations on vectors. Give the resulting polynomial.
1.
(a)
(b)
(c)
(d)
2. to . Does it correspond to a linear transformation
(a) Consider the operation on that takes
from to ? If so, what is its matrix?
