P3 = {All functions of the form p(x) = ao + a1x + a2x² + a3x³}. Maybe you remember from calculus or another algebra class that x* is called a monomial. The polynomial vector spaces are the just a span of monomials. Specifically, Pa = span{1, x, x², ..., x“}. The dimension of this space is d+1 because of the number of monomials in the above space is d +1 and the set is linearly independent. There is also a set containing every polynomial. It can be written like this. d All functions p(x) where there is a whole number d so that p(x) = > `akx* k=0
P3 = {All functions of the form p(x) = ao + a1x + a2x² + a3x³}. Maybe you remember from calculus or another algebra class that x* is called a monomial. The polynomial vector spaces are the just a span of monomials. Specifically, Pa = span{1, x, x², ..., x“}. The dimension of this space is d+1 because of the number of monomials in the above space is d +1 and the set is linearly independent. There is also a set containing every polynomial. It can be written like this. d All functions p(x) where there is a whole number d so that p(x) = > `akx* k=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The whole number d can take any value in that definition. Remember that the dimension
is number of coordinates required to represent vectors in a vector space. What is the
dimension of P? (Hint: The answer is quite large. Also, the explanation for this answer
is stated within this question. Write it in your own words!)
Transcribed Image Text:P3 = {All functions of the form p(x) = ao + a1x + a2x² + a3x³}.
Maybe you remember from calculus or another algebra class that x* is called a monomial.
The polynomial vector spaces are the just a span of monomials. Specifically,
Pa = span{1, x, x², ..., x“}.
The dimension of this space is d+1 because of the number of monomials in the above
space is d +1 and the set is linearly independent. There is also a set containing every
polynomial. It can be written like this.
d
All functions p(x) where there is a whole number d so that p(x) = > `akx*
k=0
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