M and N: Relationship in Spanning and Subsets of Polynomials

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In summary, M and N have a relationship in spanning and subsets of polynomials where M is a subset of N. They also have a close relationship in terms of spanning sets, where each set is a spanning set of the other. M can also be a proper subset of N in terms of spanning sets, but both sets are still able to generate every possible polynomial within their given parameters. While M and N may differ in terms of their subsets of polynomials, they still share the same relationship as spanning sets.
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* If p1,p2,……pm span Pn, write down a mathematical relationship between m and n.

I know that Pn means the space of all polynomials of degree at most n, and this is an (n+1) dimension space, but I am not sure what kind of mathematical relationship the question is looking for :s

Any help is greatly appreciated!
 
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If Pn has dimension n+1, what must be true of any set that spans it? (In particular how many vectors are there in any basis?)
 

1. What is the relationship between M and N in spanning and subsets of polynomials?

The relationship between M and N in spanning and subsets of polynomials is that M is a subset of N. This means that all elements of M are also elements of N, but N may have additional elements that are not in M.

2. How do M and N relate to each other in terms of spanning sets?

M and N are closely related in terms of spanning sets. In fact, M is a spanning set of N if and only if N is a spanning set of M. This means that if all elements of M can be used to create every element in N, and vice versa, then they are both spanning sets of each other.

3. Can M be a proper subset of N in terms of spanning sets?

Yes, M can be a proper subset of N in terms of spanning sets. This means that there are elements in N that cannot be created using elements from M. However, all elements in M can still be used to create every element in N.

4. What is the significance of M and N being spanning sets?

The significance of M and N being spanning sets is that they are able to generate every possible element in the set of polynomials. This means that they are a complete set of building blocks for creating all polynomials within the given parameters.

5. How do M and N differ in terms of their subsets of polynomials?

M and N may differ in terms of their subsets of polynomials in two ways. First, M may be a proper subset of N, meaning that it has fewer elements. Second, the elements in M and N may differ in terms of degree, coefficients, or other characteristics. However, they still share the same relationship in terms of being spanning sets.

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