Dimension Recovery of $S \subset P_n(F)$

Question

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?

Added from answer posted by Trancot on 18 Apr 2013:

This is what I had:

Let $S=\{f \in P_n(F) : f(1)=0\}$. Clearly, the polynomial $f(x)=0 \in S$ because $f(c)=0$ for any choice of $c\in F$. To demonstrate closure under addition and multiplication consider the fact that $cf(1)+g(1)=c\cdot 0+0=(cf+g)(1)=0$ for $f,g\in S$

Does this suffice to show subspace existence?

I know that the set $S={f \in P_n(F) : f(1)=0}$ being a subspace of $P_n(F)$ has dimension $\leq n+1$. — Trancot, Apr 17 '13 at 06:19
So... you're asking how to find the kernel of a linear transformation? (If the answer is no, then you should figure out why you should be!) — , Apr 17 '13 at 17:01
People seem to like Wikipedia these days... Here is a link of interest to those onlookers who are curious. — Trancot, Apr 17 '13 at 17:35
I'm aware. Maybe it would help to write the problem in terms of a basis for $P_n(F)$? — , Apr 17 '13 at 17:36
@Hurkyl The basis for $P_n(F)$ is ${1,x^2,x^3,\dots,x^n}$. Hmm... Can $S$ have a $1$ in its basis? — Trancot, Apr 17 '13 at 17:41

score 2 · Accepted Answer · edited Apr 18 '13 at 00:21

2

Here is one way to show this: evaluation at $1$ is a linear transformation from $P_n(F)$ to $F$. Write $\epsilon: P_n \to F$ for $\epsilon(f(x)) = f(1)$. If $f(x), g(x) \in P_n(F)$ and $k \in F$, then $\epsilon(f(x) + kg(x)) = \epsilon(f(x)) + k\epsilon(g(x)) = f(1) + kg(1)$.

The subspace of polynomials vanishing at $1$ is the kernel of this linear transformation. It is plain that $\epsilon$ is surjective, so $\dim(\text{Image}(\epsilon)) = \dim(F) = 1$. By the rank-nullity theorem, $\dim(\ker(\epsilon)) = \dim(P_n(F)) - 1 = n$.

edited Apr 18 '13 at 00:21

Trancot

4,021

answered Apr 18 '13 at 00:00

Adam Saltz

2,596

I like this, and I blindly see it as true. I don't know that there are any holes. – Trancot Apr 18 '13 at 00:08
BTW, I hope your friends and family are OK. I mean, with what happened recently in Boston and all. – Trancot Apr 18 '13 at 00:09

Dimension Recovery of $S \subset P_n(F)$

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