My intuition tells me it is.
But in terms of vectors, the span of a vector with only one component (a vector in $\mathbb{R}^1$) is not said to be a subspace of $\mathbb{R}^2$
My intuition tells me it is.
But in terms of vectors, the span of a vector with only one component (a vector in $\mathbb{R}^1$) is not said to be a subspace of $\mathbb{R}^2$
Technically, the answer is no, because $\mathbb{R}^1$ is not a subset of $\mathbb{R}^2$. $\mathbb{R}^1$ consists of real numbers, whereas $\mathbb{R}^2$ consists of ordered pairs of real numbers; therefore $\mathbb{R}^1$ is not contained in $\mathbb{R}^2$.
However, there are many ways one can "put a copy of $\mathbb{R}^1$ into $\mathbb{R}^2$", and depending on how this is done, the result may or may not be a subspace of $\mathbb{R}^2$. Look at the definition of a subspace, and consider that $S=\{(x,0)\in\mathbb{R}^2\mid x\in\mathbb{R}^1\}$ is a subspace of $\mathbb{R}^2$ because
while
$T=\{(x,1)\in\mathbb{R}^2\mid x\in\mathbb{R}^1\}$
completely fails to be a subspace of $\mathbb{R}^2$ - every condition is false:
Generally, you should note that it is entirely possible for a subspace of a vector space $V$ to have smaller dimension than $V$; for example, for any vector space $V$, the subset consisting of only the zero element is always a subspace, and it has dimension 0.