Exercise I.C.5

Question

Show that $P_n = \{$ polynomials with real coefficients of degree $\leq n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.

Toğrul Topal · Accepted Answer

\begin{itemize}
    \item $P_n$ is a vector space, since the sum of two polynomials of degree $\leq n$ cannot magically exceed $n$.
    \item By definition, every polynomial can be written in the form $\sum_{i=0}^{n}a_ix^i$, i.e., as a linear combination of the $n+1$ base polynomials $p_0(x) = x^0,p_1(x) = x^1,\ldots, p_n(x) = x^n$. Thus, $p_0,p_1,\ldots,p_n$ spans the set of all polynomials of degree $n$. Furthermore, these polynomials are linearly independent, since there cannot be a single non-trivial collection $a_0,a_1,\ldots,a_n$ such that
      $$ \forall x \in \mathbb{K}: a_0x^0 + a_1x^1 + \ldots + a_nx^n = 0$$
      (suffices to verify this for $x=1$ and $x=2$ for example)
    \end{itemize}

Exercise I.C.5

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Exercise I.C.5

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