Show that every subspace of R2 is one of the following: the trivial subspace {0}, a line through the origin, or R2 itself.
I know that all three are possible and know how to show a computational example of each, but how can I prove them? I'm new to proofs and wanted to know how to do it elegantly. Thanks!
If the only vector in the subspace $V$ is $0$, you have case 1.
If there is at least one non-zero vector $u$. Now there are two sub cases
All the vectors in $V$ are proportional to $u$. Then $V$ is the line through $u$ ( and $0$), so case 2.
There exists another vector $v$ in $V$ not proportional to $u$. Now you must show that $V$ equals $\mathbb{R}^2$. For this, show that every vector $w$ in $\mathbb{R}^2$ is a linear combination of $u$ and $v$, that is $w= a u + b v$. This translates into a system of equations for $a$, $b$, that has a (unique) solution ( consider coordinates and solve with Cramer rule the $2\times 2$ system. It has a non-zero determinant. Why?)