Vector Spaces and Subspaces

A vector space is simply a collection of things you can add together and multiply by numbers, where the results always land back in the collection. The name is a little misleading: "vector" here doesn't have to mean an arrow. It means anything that adds and scales the way arrows do, and "space" just means the whole collection of them. A subspace is a smaller vector space sitting inside a bigger one, and the test above is how you spot one. All we're really saying with it is three small checks: $\vec 0 \in H$ (the zero vector is in the set), $\vec u + \vec v \in H$ (adding two members keeps you inside) and $c\,\vec u \in H$ (scaling a member keeps you inside). It sounds abstract, but it boils down to "can I add and scale without falling out?"; bear with it.

Examples of vector spaces. You've been working in the first one all along, and the rest add and scale the same way:

• $\mathbb R^n$.
• $\mathbb P_n$: polynomials of degree at most $n$, added and scaled as usual.
• $M_{m\times n}$: all $m\times n$ matrices.
• Continuous functions on an interval; the solutions of a linear homogeneous differential equation.

That last one is superposition in disguise: add two solutions, or scale one, and you get another solution. There's an animation further down that shows this with $\sin x$ and $\cos x$.

Subspaces need only the three checks above. Think about why that's enough. Rules like $\vec u + \vec v = \vec v + \vec u$ already hold in the bigger space, so they automatically hold for the members of $H$ too; the other rules are inherited. The only thing that can go wrong is falling out of $H$, and that's exactly what the three checks catch.

A common mistake. It's tempting to think any set described by a tidy equation is a subspace, because it looks just as clean as the examples above. It isn't: the polynomials with $p(0) = 1$ miss the zero polynomial, which has $p(0) = 0$, so they fail right away. Watch out for sets that aren't closed under scaling too. The vectors with nonnegative entries survive addition, but multiply one by $-1$ and you're out.

Here's the pay-off: a theorem proved once, for "vector spaces," instantly holds for polynomials, matrices and differential equations.

Signal processing treats audio clips as vectors in a function space, and quantum mechanics treats states as vectors in (infinite-dimensional) Hilbert spaces.