TITLE: What is the point of subspaces?
QUESTION [11 upvotes]: I've been studying for my linear algebra final for the past few days, and just had this thought. 
I know what a subspace is; It's a subset of a vector space that contains the zero vector, and preserves addition and scalar multiplication. But what can we do with a set after we've discovered that it's a subspace? The questions in my textbook only go as far as telling us to determine if a set is a subspace or not, we're not asked to use the fact that a set is a subspace to do something else. So my question is, what is the point of a subspace then?
Thanks

REPLY [2 votes]: As you probably know, the three-dimensional space $\mathbb{R}^3$ in which we all live is a vector space. The linear subspaces of this space are lines and planes that pass through the origin. If you're trying to do anything with a geometric flavor to it, then lines and planes are important, of course. Actually, lines and planes that do not pass through the origin are useful, too. These are not linear subspaces, they are affine subspaces, which are related.
So, what can you do with lines and planes that you can't do with arbitrary subsets of $\mathbb{R}^3$? I'm sure you can think of some things. Here are a few examples:

You can easily parameterise them. A line is a set of vectors of the form $t\mathbf{a}$, where $\mathbf{a}$ is some vector and $t$ is a number. Similarly, a plane is a set of vectors of the form $u\mathbf{a} + v\mathbf{b}$.
You can measure distance to a line or plane easily (just by finding a perpendicular vector).
They're easy to intersect. The intersection of two planes through the origin is either a line or a plane. Both are subspaces.
Lines and planes give us concepts of tangency.