## Linear independence of polynomials

One of the exercises this week asked for a proof of linear independence for the set

inside the polynomials with real coefficients. However, note that the polynomials here are regarded as *functions* from to . Thus, it amounts to showing that if

as a function, then all have to be zero. This does require proof. One quick way to do this is to note that all polynomial functions are differentiable. And if

is the zero function, then so are all its derivatives. In particular,

for all . But Thus, for all .

One possible reason for confusion is that there is another ‘formal’ definition of by simply identifying a polynomial with its sequence of coefficients. That is, you can think of an element of as a function that has *finite support* in that for all but finitely many . With this definition, the polynomial becomes identified with the function that sends to 1 and everything else to zero. If you take this approach, the linear independence also becomes formal. But in this problem, you are defining as a function in its variable. This of course is the natural definition you’ve been familiar with at least since secondary school.

Here are two questions:

1. If you think of two polynomials and as functions from to with finite support, what is a nice way to write the product ?

2. What is the advantage of this formal definition?

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