What is {p(t):p(2) = 0} in linear algebra? In my linear algebra homework, it asks me if the subset of p2 is the subspace of p2, like the question no.2. I know how to verify if it’s the subspace, but I don’t get what it means by {p(t):p(2) = 0}.
From the context I would say $\{p(t) \,:\, p(2)=0\}$ are those polynomials that vanish at $t=2$, meaning if you substitute $t=2$ the polynomial evaluates to $0$. You could also say that this set is the set of polynomials which have $t=2$ as a zero/root.
$P_2$ is the set of all polynomial functions of degree at most 2

i.e. $p(t) = a t^2 +b t + c$

$\{p(t):p(2)=0\}$ would be the subset such $p(2)=0$

$p(t) = a (t-2)^2 + b(t-2) = at^2 + (-2a+b)t + 4a-2b$

What is the degree, what is the basis?