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
$\endgroup$ 25 Answers
$\begingroup$It is a very rare situation in mathematics to find oneself looking at a particular set and thinking, "hmm, I wonder whether this is a subspace".
In fact, this just about only happens when you're in a beginning linear algebra course and doing exercises. And the point of those exercises is not to train the skill of telling subspaces from non-subspaces (which in itself is pretty useless), but to give you an opportunity to develop an intuition for what a subspace is, and in particular what you can depend on if someone else gives you a set and promises, "oh, by the way, I've made sure this is a subspace".
A lot of subsequent concepts and results in linear algebra are formulated in terms of subspaces, so in order to properly understand and internalize those results one needs to be absolutely familiar with the idea of what a subspace is -- not just on the level of being able to reproduce the definition on command, but on the level of having an immediate idea what can and cannot be expected of a subspace.
For example, in order to speak about dimension you need a subspace. If you have a system of linear equations, the solution set is a translated subspace. Eigenspaces are subspaces. Spans are subspaces. Orthogonal complements are subspaces. If you have a linear transformation between vector spaces, the image of a subspace is a subspace, and the preimage of a subspace is also a subspace -- in particular, the image of the entire domain is a subspace, and the preimage of $\{0\}$ is a subspace. This alone can teach you a lot about the structure of sets in a vector space where concrete calculations may be difficult to carry out or even imagine intuitively.
$\endgroup$ $\begingroup$The set of all solutions to a linear homogeneous differential equation: $$a_n(x)\frac{d^ny}{dx^n}+ a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+ \dots+ a_1(x)\frac{dy}{dx}+ a_0(x)y= 0$$ form an $n$ dimensional subspace of the space of all $n$ times differentiable functions. That means that means that, while there are an infinite number of functions satisfying that equation, if we can find just $n$ independent solutions, $y_1(x),$ $y_2(x),$ $\dots,$ $y_{n-1}(x),$ $y_n(x),$ then we can write any solution, $y(x),$ as $$y(x)= C_1y_x(x)+ C_2y_2(x)+ \dots+ C_1y_1(x).$$ The crucial point is that the definition of "subspace" the sum of two such functions or such a function times a number is still in the subspace.
The set of all solution to the linear non-homogeneous differential equation, $$a_n(x)\frac{d^ny}{dx^n}+ a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+ \dots+ a_1(x)\frac{dy}{dx}+ a_0(x)y= f(x)$$ do not form a subspace so we cannot form new solutions by adding old ones or multiplying by numbers.
$\endgroup$ $\begingroup$It matters for the same reason why it matters that something is a vector space: you have the entire theory of linear algebra at your disposal.
$\endgroup$ 2 $\begingroup$Think of one spherical body rotating around another under the influence of their mutual gravitational attraction. It's not, I think, trivial or uninteresting to prove that the movement of the bodies' centers is in a plane, i.e., a 2-dimensional subspace of three-dimensional space. Similarly in chemistry it is not trivial that benzene is planar (i.e., the centers of the nuclei of its carbons lie in a 2-dimensional subspace) whereas cyclohexane is not.
$\endgroup$ $\begingroup$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.
