Why is $\mathbb{R}$ not a linear subspace of $\mathbb{R}^3$? I thought that $\mathbb{R}$ would just be any 1-dimensional line in $\mathbb{R}^3$ and that as long as you multiply or add two vectors on that line together, you'd still be in $\mathbb{R}$.

Every one-dimensional subspace of $\mathbb R^3$ is isomorphic to $\mathbb R$ as $\mathbb R$ vector spaces. For many people, isomorphism is enough to call that copy "$\mathbb R$", but one has to remember that it isn't special and that there are many other such copies.
Someone might object to saying that $\mathbb R$ is a subspace if they want to say that it isn't even a subset: $\mathbb R^3$ consists of triples of real numbers while $\mathbb R$ are "1-tuples" of real numbers.
