Answer
See below
Work Step by Step
Given: $S =\{(a, 2^a) : a \in R\}$ and the vector space $V$ given in Problem $13$.
Take $(1,2^1)=(1,2) \in W$. We can see $W$ is nonempty (1)
Let $v=(a,2^a)\\
w=(b,2^b)$
then $v+w=(a,2^a)+(b,2^b)=(a+b,2^a+2^b)=(a+b,2^{a+b})\\
\rightarrow v+w \in S$
Hence, $v+w$ is closed under addition multiplication (2)
Let $k$ be a scalar
Obtain $kv=(ka,(2^a)^k)=(ka,2^{ka})\\
\rightarrow kv \in S$
Hence, $kv$ is closed under scalar multiplication (3)
From (1)(2)(3), $S$ is a subspace of $V$
