Answer
proof: please see step-by-step
Work Step by Step
see Theroem 1, p.196:
If $v_{1},v_{2}...,v_{p}$ are in a vector space $V$, then
Span $\{v_{1},v_{2}...,v_{p}\}$ is a subspace of V.
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Let $f(t)=t^{2}$. Then, $f(t)\in \mathbb{P}_{n}$ (for $n \geq 2)$
and
$W=\{p(t)| p(t)=a\cdot f(t), a\in R\} =$ Span $\{f(t)\}$
by Th.1, $W$ is a subspace of $\mathbb{P}_{n}$.