# Subspaces

Sorry, today’s post is kind of lame, but for now, I would rather keep them short while I have the time to post pretty much daily.

I’ve been saying that you should be thinking about $\mathbb R^2$ and $\mathbb R^3$ as examples, since they are familiar. You may have noticed that there are copies of $\mathbb R^2$ inside $\mathbb R^3$. The $xy$-plane is one such example. The $xz$-plane, and the $yz$-plane area also copies of $\mathbb R^2$ inside $\mathbb R^3$. We call these subspaces for the obvious reason.

$W$ is a subspace of a vector space $V$ if $W\subseteq V$ and $W$ is also a vector space.

So $\{0\}$ is a (slightly boring) subspace of any vector space. Also, a vector space $V$ is a subspace of itself. But we can get crazier. $\mathbb C$ is a vector space, right? Well yes, if I tell you what the scalars are. They can be in any field, but for simplicity, think of them as in $\mathbb R$. So its pretty much just the plane, like $\mathbb R^2$. The real line $\mathbb R$ is also a vector space over the field $\mathbb R$, so it is a subspace of $\mathbb C$.

One last point. subspaces don’t have to be “pointing in some cardinal direction.” In fact, this has no meaning, but all of the examples I gave so far have that property. Subspaces can come in at “weird angles.” So here’s another example:

$\{a\cdot \pi+a\cdot i\mid a\in\mathbb R\}$

is a subspace of $\mathbb C$. It’s clearly a subset of $\mathbb C$. You can verify that it is also a vector space.

As an aside, most new algebraic objects I discuss, I’ll talk about the objects, subobjects, homomorphisms, and quotient objects. Subspaces are the subobjects of vector spaces. Eventually, we’ll talk about linear transformations (the homomorphisms), and quotient vector spaces. These are common ideas in algebra, and many of the theorems about one algebraic object have analogous theorems in another “category.”