Prove that there exists an element of V which cannot be expressed as a linear combination of elements of X.

Question

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Let V = P(R) be the vector space of polynomials with coefficients in R and let X be a finite subset of V . Prove that there exists an element of V which cannot be expressed as a linear combination of elements of X.

Answer 1

Hint:

Since $X$ is finite, there is $n\in\Bbb N$ such that $n>\deg(x),\,\forall x\in X$. And note that $\deg(x+y)\leq\max\left\{\deg(x),\deg(y)\right\}$.