Exercise 1.4.14

For $v\in \mathrm{span}(S_1\cap S_2)$ we may write $v={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{a}_i{x}_i$ with $x_i\in S_1$ and $x_i\in S_2$. So $v$ is an element of both $\mathrm{span}(S_1)$ and $\mathrm{span}(S_2)$ and hence an element of $\mathrm{span}(S_1)\cap \mathrm{span}(S_2)$. For example we have if $S_1=S_2=(1,0)$ then they are the same and if $S_1=(1,0)$ and $S_2=(0,1)$ then we have the left hand side is the set of zero vector and the right hand side is the the plane ${ℝ}^2$.