Given: $$V = P^4$$ That is, $V$ is the set of all polynomials in $x$ of degree at most four.
If $S = \{1+x^2, x^3, 2x^3\}$
Can: $$S \subset V \space\space ?$$ I would say yes based strictly on the definition of subsets.
Questions:
Is $\text{span}(S)$ a subspace of $V$?
I understand that in order to be a subspace, the subset needs to be closed under addition and scalar multiplication. Given this, is $S$ also a subspace of $V$? Because you will never leave the bounds of $V$?
Is $S$ a basis of $V$?
I would say no, since not only is $S$ not linearly independent, it does not span $V$. That is, it is missing terms like $x$ and $x^4$.
Can you make a basis by including elements of $S$ + additional terms? Or should the basis simply be $\{1, x, x^2, x^3, x^4\}$?
If you can make a basis by including elements of $S$, would the elements be $\{1+x^2, x^3\} \cup \space \{1, x, x^4\}$?
Thank you for the clarification!
We have that
$S$ is a subset of V but not a subspace (the span of $S$ is a subspace)
$S$ can't be a basis of $V$ because a basis must contain exactly $5$ linearly independent elements
yes, including $1$, $x^4$ and $x$ and eliminating $x^3$ or $2x^3$ it becomes a basis