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If $p_1,p_2,...,p_k$ are linearly independent polynomials in $P_n$, a mathematical relationship between $k$ and $n$ is:

$k\le n.$

If the k will be more than n, the set of polynomials cannot be linearly independent, is it correct?

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If by $P_n$ you mean the polynomials of degree $\le n$, then your relation should be $k \le n+1$: for example, $1 (= x^0), x^1, \ldots, x^n$ is a basis. It follows that a set of more than $n+1$ members of $P_n$ cannot be linearly independent.

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