Infinite Basis for Polynomials I read the example and understood most of it,but
ID: 3116574 • Letter: I
Question
Infinite Basis for Polynomials
I read the example and understood most of it,but have some questions to ask:
1.Is the first highlighted part a mistype?I thought the basis would be a infinite set of fk(x)
where fk(x)=x^k?
2.The last part starting from "a final remark",It said that infinite bases have nothing to do with infinte linear combinations.
What does that mean ?What are infinte linear combinations?
3.Continue reading,the last part said that readers who inject power series into this example should read again.
I thought polynomials can be written in the form of power series, so whats the difference?
why shouldn't we put power series into this example?
Bases and Dimension Sec. 2.3 ce EXAMPLE 16. We shall now give an example of an infinite basis. Let F be a subfield of the complex numbers and let V be the space of poly- nomial functions over . Recall that these functions are the functions from P into F which have a rule of the form LetA(z) = zk, k = 0, 1, 2, . . . . The (infinite) set {fJaf, . . .} is a basis for V. Clearly the set spans V, because the function f (above) is The reader should see that this is virtually a repetition of the definition of polynomial function, that is, a function f from F into F is a polynomial function if and only if there exists an integer n and scalars co,.., c, such that -coa Why are the functions independent? To show that the set o is independent means to show that each finite subset of it is independent. It will suffice to show that, for each n, the set fof is independent. Suppose that This says that for every z in F; in other words, every z in F is a root of the polynomial f(x) = co + c1x + + c.xn. we assume that the reader knows that a polynomial of degree n with complex coefients cannot have more than n distinct roots. It follows that co-c'. . . . = c" 0. We have exhibited an infinite basis for V. Does that mean that V is not finite-dimensional? As a matter of fact it does; however, that is not immediate from the definition, because for all we know V might also have a finite basis. That possibility is easily eliminated. (We shall eliminate it in general in the next theorem.) Suppose that we have a finite number of polynomial functions g,.. gr. There will be a largest power of z which There appears (with non-zero coefficient) in g(c)g(x). If that power is k clearly fk+1(x)-z"+1 is not in the linear span of gi, . . . , gr. So V is not finite-dimensional A final remark about this example is in order. Infinite bases have nothing to do with 'infinite linear combinations.' The reader who feels an irresistible urge to inject power series into this example should study the example carefully again. If that does not efect a cure, he should consider restricting his attention to finite- dimensional spaces from now on.Explanation / Answer
1.Is the first highlighted part a mistype?I thought the basis would be a infinite set of fk(x)
where fk(x)=x^k?
Answer:
YES, You ar thinking right. That should be x^k for kth degree polynomial.
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2.The last part starting from "a final remark",It said that infinite bases have nothing to do with infinte linear combinations.
What does that mean ?What are infinte linear combinations?
Answer:
We know that for same set, there can be more than one bases.
So that means for the given infinite set we can have infinite different bases.
But for our study or calculation one basis is sufficient.
So basically "infinite bases have nothing to do with infinte linear combinations." means you don't have to worry about finding infinite bases for given infinite set.
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3.Continue reading,the last part said that readers who inject power series into this example should read again.
I thought polynomials can be written in the form of power series, so whats the difference?
Answer:
I totally agree with you that polynomials can be written in the form of power series. I think they just want to say that yes you can use power series or you may find more information by reading next page.