- #1

Mr Davis 97

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## Homework Statement

Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent.

## Homework Equations

## The Attempt at a Solution

We will proceed by contraposition.

Assume that S is a linearly dependent set. Thus there exists a linear dependence relation ##a_1p_1 + \cdots + a_np_n = 0## such that ##a_1,...,a_n## are not all zero. Now, the RHS of the relation has degree -1. Hence, the LHS must have degree -1 also. In this case, assume that no two polynomials have the same degree. Then the LHS would have the degree of the polynomial with largest degree, which is other than -1 since there are no zero polynomials in the set S. This is a contradiction, because the LHS must have degree -1. Hence, there must exist at least two polynomials with the same degree.

In this correct? Am I begging the question if I claim that the LHS must have degree -1 also?