I'm currently working on a 3 step problem which goes as follows:
a) Prove that the only subspace of $\mathbb{R}^1$ are the zero subspace and $\mathbb{R}^1$.
Proof: Let $V$ be a subspace of $\mathbb{R}^1$ such that $V\neq\mathbb{R}^1$ and $V\neq\{0\}$, then there exist some $x\in \mathbb{R}^1$ such that $x\notin V$. Let $v$ be the vector $(v_1)$ such that $v_1\neq0 $. Since $V$ is not the zero space, it is evident that such an element exists. Then construct the vector $$v'=\frac{x}{v_1} v=(x)$$ But we already determined that $x\notin V$, so $v'\notin V$. Therefore, $V$ can't be a subspace of $\mathbb{R}^1$ if $V\neq\mathbb{R}^1$ and $V\neq\{0\}$.
b) Prove that a subspace of $\mathbb{R}^2$ is $\mathbb{R}^2$, the zero space or a line through the origin.
Here I'm starting to have some trouble. The book would like me to solve this in much the same way as I solved a), but I was unable to set up a solid proof with that method. Instead I turned to the dimensions of linearly independent vectors.
Proof: By the rules of subspaces, if $W$ is a subspace and $\alpha \in W$, then $c\cdot\alpha\in W$. Because of this, any subspace $V$ of $\mathbb{R}^2$ is at least a line (see a)). Let $W$ denote this line, and $v\in V$ such that $v \notin W$. Let us write $v=(x,y)$ and pick some vector $w=(a,b)\in W$. Then $\nexists c\in\mathbb{R}: c\cdot w=v$. Therefore, $v$ and $w$ are linearly independent vectors which means that the span a two dimensional space. Because $\mathbb{R}^2$ is two dimensional, that means that any subspace of $\mathbb{R}^2$ that is not either a $\mathbb{R}^2$ or the zero space must necessarily be a line.
c) Describe all the subspaces of $\mathbb{R}^3$.
It is clear that these are all either $\mathbb{R}^3$, the zero space, a line through the origin, or a plane through the origin. But proving this (for me) would result in an argument that is completely analogous to the proof I gave in b). So I was wondering if someone could explain to me a simpler method of doing such a proof if one assumes no prior knowledge of bases and dimensions?
Thanks in advance!