Professor Dave here, let’s talk subspaces.
Once we understand what makes up a vector space, it is fairly easy to understand the
concept of subspaces as well.
To put it simply, a subspace is a smaller set within a vector space that is itself a
vector space.
Let’s look at some examples to clarify.
Let’s say that V is a vector space.
Now we can highlight this smaller section of V, and call it S. Every element of S was
already also in our vector space V, so we know they all obey the properties we discussed
in the previous tutorial.
This means that the only requirement we need to check in order to determine whether S is
a vector space, is that it satisfies the properties of closure.
If S is closed, it will also be a vector space, and because it is completely contained in
the larger vector space V, we will then call it a subspace of V.
We saw last time that R3, or the set of all real vectors of length three, was a vector space.
So now let’s look at a specific choice of vectors contained within R3 and see if it
makes up its own vector space, thus qualifying as a subspace of R3.
For this example we will choose a set S made of vectors that have the form x, 0, -x.
By limiting the form of the vector like this, we have chosen a smaller set of R3 while knowing
that our chosen set of vectors still follows the rules of the larger set.
What we must check now is if our smaller set is closed, by checking the two properties
of closure.
First let’s multiply a vector x by a scalar c.
Distributing the scalar c, we end up with a vector cx, 0, -cx.
Notice that this still has the same form that we want for our set, with some value on the
first row, 0 for the middle row, and the negative version of the first value in the third row.
This vector is therefore still in our set S. Next, let’s take two vectors x and y,
both from the set S, and add them together.
Provided that we factor the negative sign out of this sum, we will see that the third
entry is simply the negative version of the first, and once again we find a result in
the same form as we started.
This means that the sum is also contained in our set S. We have verified that S is closed,
making it a vector space.
And because it was contained within the larger vector space V, this makes S a subspace of V.
Before going further, it’s important to
introduce a few more concepts.
Once again let’s say we have a vector space V, and a number of elements of V, which we
can call v1, v2, all the way to vn.
Any sum of these elements multiplied by some scalars, which we can represent as a1v1 plus
a2v2 and so on, up to anvn is called a linear combination.
The set of all possible linear combinations of these elements is called their span.
As an example, let’s take three vectors from R3, those being 2, 1, -1; 0, 2, 2; and
-1, -1, -1.
The span of these three vectors will be given by the linear combination av1 + bv2 + cv3
where a, b, and c are all scalars.
Let’s plug in our vectors and distribute the scalars, and then do vector addition,
to see that the span of these three vectors has the form 2a - c, a + 2b - c, -a + 2b - c.
It’s worth noting that the span of any number of elements of a vector space V, is also a
subspace of V.
In fact, it is the smallest subspace of V that contains that set of elements, as it
is the intersection of all subspaces that contain them.
Span will end up playing a significant role when it comes to completely describing vector
spaces, but for now, let’s check comprehension.