We spent a good deal of time on the idea of a null space.
What I'm going to do in this video is introduce you to a
new type of space that can be defined around a matrix, it's
called a column space.
And you could probably guess what it means just based on
what it's called.
But let's say I have some matrix A.
Let's say it's an m by n matrix.
So I can write my matrix A and we've seen this multiple
times, I can write it as a collection of columns vectors.
So this first one, second one, and I'll have n of them.
How do I know that I have n of them?
Because I have n columns.
And each of these column vectors, we're going to have
how many components?
So v1, v2, all the way to vn.
This matrix has m rows.
So each of these guys are going to have m components.
So they're all members of Rm.
So the column space is defined as all of the possible linear
combinations of these columns vectors.
So the column space of A, this is my matrix A, the column
space of that is all the linear combinations of these
column vectors.
What's all of the linear
combinations of a set of vectors?
It's the span of those vectors.
So it's the span of vector 1, vector 2, all the
way to vector n.
And we've done it before when we first talked
about span and subspaces.
But it's pretty easy to show that the span of any set of
vectors is a legitimate subspace.
It definitely contains the 0 vector.
If you multiply all of these guys by 0, which is a valid
linear combination added up, you'll see that it contains
the 0 vector.
If, let's say that I have some vector a that is a member of
the column space of a.
That means it can be represented as some linear
combination.
So a is equal to c1 times vector 1, plus c2 times vector
2, all the way to Cn times vector n.
Now, the question is, is this closed under multiplication?
If I multiply a times some new-- let me say I multiply it
times some scale or s, I'm just picking a random letter--
so s times a, is this in my span?
Well s times a would be equal to s c1 v1 plus s c2 v2, all
the way to s Cn Vn Which is once again just a linear
combination of these column vectors.
So this Sa, would clearly be a member of the
column space of a.
And then finally, to make sure it's a valid subspace-- and
this actually doesn't apply just to column space, so this
applies to any span.
This is actually a review of what we've done the past. We
just have to make sure it's closed under addition.
So let's say a is a member of our column space.
Let's say b is also a member of our column space, or our
span of all these column vectors.
Then b could be written as b1 times v1, plus b2 times v2,
all the way to Bn times Vn.
And my question is, is a plus b a member of our span, of our
column space, the span of these vectors?
Well sure, what's a plus b? a plus b is equal to c1 plus b1
times v1, plus c2 plus v2 times v2.
I'm just literally adding this term to that
term, to get that term.
This term to this term to get this term.
And then it goes all the way to Bn and plus Cn times Vn.
Which is clearly just another linear
combination of these guys.
So this guy is definitely within the span.
It doesn't have to be unique to a matrix.
A matrix is just really just a way of writing a
set of column vectors.
So this applies to any span.
So this is clearly a valid subspace.
So the column space of a is clearly a valid subspace.
Let's think about other ways we can interpret this notion
of a column space.
Let's think about it in terms of the expression-- let me get
a good color-- if I were to multiply my-- let's think
about this.
Let's think about the set of all the values of if I take my
m by n matrix a and I multiply it by any vector x, where x is
a member of-- remember x has to be a member of Rn.
It has to have n components in order for this multiplication
to be well defined.
So x has to be a member of Rn.
Let's think about what this means.
This says, look, I can take any member, any n component
vector and multiply it by a, and I care about all of the
possible products that this could equal, all the possible
values of Ax, when I can pick and choose any
possible x from Rn.
Let's think about what that means.
If I write a like that, and if I write x like this-- let me
write it a little bit better, let me write x like this-- x1,
x2, all the way to Xn.
What is Ax?
Well Ax could be rewritten as x1-- and we've seen this
before-- Ax is equal to x1 times v1 plus x2 times v2, all
the way to plus Xn times Vn.
We've seen this multiple times.
This comes out of our definition of
matrix vector products.
Now if Ax is equal to this, and I'm essentially saying
that I can pick any vector x in Rn, I'm saying that I can
pick all possible values of the entries here, all possible
real values and all possible combinations of them.
So what is this equal to?
What is the set of all possible?
So I could rewrite this statement here as the set of
all possible x1 v1 plus x2 v2 all the way to Xn Vn, where
x1, x2, all the way to Xn, are a member of the real numbers.
That's all I'm saying here.
This statement is the equivalent of this.
When I say that the vector x can be any member of Rn, I'm
saying that its components can be any
members of the real numbers.
So if I just take the set of all of the, essentially, the
combinations of these column vectors where their real
numbers, where their coefficients, are members of
the real numbers.
What am I doing?
This is all the possible linear combinations of the
column vectors of a.
So this is equal to the span v1 v2, all the way to Vn,
which is the exact same thing as the column space of A.
So the column space of A, you could say what are all of the
possible vectors, or the set of all vectors I can create by
taking linear combinations of these guys, or the span of
these guys.
Or you can view it as, what are all of the possible values
that Ax can take on if x is a member of Rn?
So let's think about it this way.
Let's say that I were to tell you that I need to solve the
equation Ax is equal to-- well the convention is to write a b
there-- but let me put a special b
there, let me put b1.
Let's say that I need to solve this equation
Ax is equal to b1.
And then I were to tell you-- let's say that I were to
figure out the column space of A-- and I say b1 is not a
member of the column space of A So what does that tell me?
That tells me that this right here can never take on the
value b1 because all of the values that this can take on
is the column space of A.
So if b1 is not in this, it means that this cannot take on
the value of b1.
So this would imply that this equation we're trying to set
up, Ax is equal to b1, has no solution.
If it had a solution, so let's say that Ax equals b2 has at
least one solution.
What does this mean?
Well, that means that this, for a particular x or maybe
for many different x's, you can definitely
achieve this value.
For there are some x's that when you multiply it by a, you
definitely are able to get this value.
So this implies that b2 is definitely a member of the
column space of A.
Some of this stuff on some level it's almost obvious.
This comes out of the definition
of the column space.
The column space is all of the linear combinations of the
column vectors, which another interpretation is all of the
values that Ax can take on.
So if I try to set Ax to some value that it can't take on,
clearly I'm not going to have some solution.
If I am able to find a solution, I am able to find
some x value where Ax is equal to b2, then b2 definitely is
one of the values that Ax can take on.
Anyway, I think I'll leave you there.
Now that you have at least a kind of abstract understanding
of what a column space is.
In the next couple of videos I'm going to try to bring
everything together of what we know about column spaces, and
null spaces, and whatever else to kind of understand a matrix
and a matrix vector product from every possible direction.