Problem
Is $\mathbb{R}^2$ a subspace of the complex vector space $\mathbb{C}^2$?
Solution
No, $\mathbb{R}^2$ is not a subspace of $\mathbb{C}^2$ - a subspace has the same addition and scalar multiplication as the space it’s a subspace of - see definition 1.32. But $\mathbb{C}^2$ is a vector space over $\mathbb{C}$, thus for any scalar $\lambda \in \mathbb{C}$ any vector $v$ of our subset $\mathbb{R}^2$ should satisfy $\lambda v \in \mathbb{R}^2$, but for $\lambda = i$, this is clearly not the case - take e.g. $(1,0) \in \mathbb{R}^2$, then $i \cdot (1,0) ; \cancel{\in} ; \mathbb{R}^2$ and $\mathbb{R}^2$ is thus not a subspace of $\mathbb{C}^2$.