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I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book hasn't explained everything properly so my question might be extremely easy, apologize in advance:

For $\mathbb{R}^n$, we defined the vectors in the standard basis like $\mathbf{e}_1:=\begin{pmatrix} 1\\ \vdots\\ 0 \end{pmatrix}_{n\times1}$ until $\mathbf{e}_n:=\begin{pmatrix} 0\\ \vdots\\ 1 \end{pmatrix}_{n\times1}$.

But for the polynomials $\mathbb{P_n}$, we defined the vectors for the standard basis like $\mathbf{e}_0:=1;\mathbf{e}_1:=t;...;\mathbf{e}_n:=t^n$ .

Since for $\mathbb{R}^n$, the vectors of the standard basis start from 1, the size of each vector is n. But for $\mathbb{P}_n$ the size of each vector is n+1 (since the vectors of the standard basis start from 0 and continue till n). If what I said is correct, then why we say $\mathbb{P}_n$ instead of $\mathbb{P}_{n+1}$ when the size of each vector is n+1? Isn't it the case that $\mathbb{R}^n$ is $\mathbb{R}^n$ because the size of each vector is n? If what I said is wrong, please point out to the mistake and any help would be appreciated!

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It seems natural to denote the polynomials of degree $\le n$ by some name like $P_n$. Yes, as you point out the space has dimension $n+1$ over the field of coefficients. We could use a name such as $P_{\lt n+1}$, but that is less attractive. –  André Nicolas Aug 10 at 16:22

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The notation $\mathbb R^n$ originally comes from the fact that the set is the $n$th power of $\mathbb R$, that is $\mathbb R^n = \underset{n \text{ times}}{\underbrace{\mathbb R\times \dots\times \mathbb R }}$. Then, since $n$ also is the dimension, the usage generalizes to things like $S^n$, the $n$-dimensional sphere.

But dimension is not the only thing we may want to emphasize. When working with polynomials, their degree is pretty important. And so, the notation for the space of polynomials of degree at most $n$ has maximal degree in it, as $\mathbb P_n$.

Using dimension to index polynomial spaces would quickly lead us in trouble when we have more than one variable. The space of polynomials of degree at most $n$ in $k$ variables has dimension $\binom{n+k}{k}$. Not something you would want to put in the sub- or superscript.

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