Answer
See answer below
Work Step by Step
We are given $S=\{(a,2^a):a \in R\}$ and $V$ is a vector space of problem 14.
Let $a=1$ we have $(1,2^1)=(1,2) \in W$. Then $W$ is non-empty.
Since $0 \lt 2^a$ we have $(a, 2^a) \in V$ for all $a \in R$ so $W \subset V$
Assume that: $v=(a,2^a) \\
w=(b,2^b) \\
v+w=(a,2^a)+(b,2^b)=(a+b,2^a\times2^b)=(a+b,2^{a+b}) \\
\rightarrow v+w \in S$
Given a scalar $k$ we have $kv=[ka,(2^a)^k]=(ka,2^{ka}) \rightarrow kv \in S$
Hence $S$ is a non-empty subset of $V$ which is closed under scalar multiplication by vectors and therefore $S$ is a subspace of $V$.
