Chapter 4 - Vector Spaces - 4.1 Exercises - Page 198: 5

proof: please see 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. ------------ 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 $\mathbb{P}_{n}$.