Column Space of a Matrix

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.