I have the vector space $V$ above that belongs to $\mathbb{F}$, and $V$ is the group of all polynomials that are of degree $3$.
$W= \{ p \in V | p(1)=p(-1)=0\}$
1.) Prove that W is a subspace of $V$.
A.) Ok let's prove that the space isn't empty.
Let's take $p(x)= 0$, and this satisfies the condition of $p(1)=p(-1)=0$
B.) Now that the sum of two vectors in $W$ belong in $V$
Let's take two polynomials that satisfy $W$ and show that their sum belongs in $W$. This is pretty trivial since you can't go increase your degree from addition.
Let's take $p_1(x)=x_1^3-x_1^2-x_1+1$ and $p_2(x)=x_2^3-x_2^2-x_2+1$.
Their sum belongs in $V$.
Let's check multiplication by a scalar $\lambda \in \mathbb{R} $(also pretty trivial):
$p(\lambda x)=\lambda x^3-\lambda x^2-\lambda x+1 = \lambda p(x)$
Now onto the basis:
My thought process is that I might need to find a basis that contains at least 4 components (since $\mathbb{R}_3[x]$ contains 4 components), which I don't think is correct.
So far I have two components:
B= $\{ (x^2-1),(x(x^2-1))\} $
This is where I'm stuck and can't find anymore polynomials that satisfy the requirement.
TL;DR
1.) Is my thought process for proving that $W$ is a subspace of $V$ correct?
2.) How can I find a basis for $W$? Is there a simpler way using a system of equations rather than brute force trial and error?
Best Answer
Sorry, but your line of thought is quite wrong.
Your part B is the worst one. Apart from using $x_1$ and $x_2$, which is already bad, you can't prove the set is closed under addition by showing that the sum of two particular element is in the set. You need to show that for any choice of polynomials $p,q\in W$, the sum $p+q\in W$. Similarly for the closure with respect to scalar multiplication.
Let's try a more systematic approach.
Consider the vector space $V$ of polynomials of degree at most $3$ (not *of degree $3$) with real coefficients.
For $a\in\mathbb{R}$, consider $W_a=\{p\in V: p(a)=0\}$. Then $W_a$ is a subspace of $V$. Indeed, the zero polynomial clearly belongs to $W_a$. Moreover, if $p,q\in W_a$, we have $p(a)=0$ and $q(a)=0$; so, if $r(x)=p(x)+q(x)$, we have $$ r(a)=p(a)+q(a)=0+0=0 $$ because of standard laws of polynomials. Finally, if $\lambda\in\mathbb{R}$ and $s(x)=\lambda p(x)$, we have $$ s(a)=\lambda p(a)=0 $$ Therefore $p(x)+q(x)\in W_a$ and $\lambda p(x)\in W_a$.
Your set is $W=W_1\cap W_{-1}$ and is a subspace being the intersection of two subspaces.
In order to find a basis, you can consider the linear map $$ f\colon V\to\mathbb{R}^2,\qquad f(p)=\begin{bmatrix}p(1)\\p(-1)\end{bmatrix} $$ and notice that $W=\ker f$. (Note that linearity of this map is basically the same as the proof above.)
It's easy to find the matrix of $f$ relative to the standard bases on $V$ (that is, $\{1,x,x^2,x^3\}$ and of $\mathbb{R}^2$. It is $$ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \end{bmatrix} $$ A standard Gaussian elimination brings to the reduced row echelon form $$ \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix} $$ and the null space of this matrix has as basis $$ \left\{ \begin{bmatrix}-1\\0\\1\\0\end{bmatrix}, \begin{bmatrix}0\\-1\\0\\1\end{bmatrix} \right\} $$ The polynomials having these coordinates with respect to the standard basis are $x^2-1$ and $x^3-x$, so a basis for $W$ is $$ \{x^2-1,x^3-x\} $$ as was your intuition.