PROFESSOR: So as long
as I'm introducing
the idea of a vector
space, I better introduce
the things that go with it.
The idea of its dimension
and, all important, the idea
of a basis for that space.
That space could be all of
three dimensional space,
the space we live in.
In that, case the
dimension is three,
but what's the meaning of
a basis-- a basis for three
dimensional space.
Or a basis for other spaces.
OK, so I have to explain
independence, basis,
and dimension.
Dimension's easy if
you get the first two.
OK, independence.
Are those vectors independent?
Well, if I draw them, in
three dimensional space,
I can imagine 2, 1, 5
going in some direction.
Let me draw it.
How's that?
2, 1, 5, whatever!
Goes there.
That's a1.
OK.
Now is a2 on the same line?
If a2 is on the same line
then it would be dependent.
The two vectors
would be dependent
if they're on the same line.
But this one is
not on that line.
A 4, 2, 0.
So it doesn't go up and all.
It's somewhere in
this plane, 4, 2, 0.
I'll say there.
Whatever.
a2.
So those are independent.
So their combinations
give me a space.
The combinations of a1 and a2
give me a plane, a flat plane,
in three dimensional space.
That plane is, I would
say, they span the plane.
a1 and a2 span a plane.
And here's the key word: span.
So there are two vectors.
They're in three
dimensional space.
And the plane they span
is all their combinations.
That's what we're always doing:
taking all the combinations
of these vectors.
OK.
So there-- and actually, a1 and
a2 are a basis for that pane.
a1 and a2 are a
basis for that plane
because their combinations
fill the plane.
And also, they're independent.
I need them both.
If I threw away one, I would
only have one vector left,
and it would only span a line.
OK.
Now let me bring in a third
vector in three dimensions.
Well, what shall I take
for that third vector?
Ha!
Suppose I take a1 plus
a2 as my third vector.
So 6, 3, 5.
What about the vector 6, 3, 5?
Well, what do I know?
It's obviously special.
It's a1 plus a2.
It's in the same plane.
So if I took a3 equal 6, 3,
5, that would be dependent.
The three vectors would
be dependent with that a3.
They would span the plane still.
Their combinations would
still give the plane,
but they wouldn't be
a basis for the plane.
a1 and 12 and a3 together,
that's too much, too many
vectors for a single plane.
The vectors are dependent.
And we don't-- a basis has
to be independent vectors.
You have to need them all.
We don't need all three here.
So that's a dependent one.
It can't go into a
basis with a1 and a2
because the three
vectors are dependent.
Now let me make a
difference choice.
So that one's dead.
That did not do it.
All right.
Let me take a3
equal to some other,
not a combination of
these, but headed off
in some new direction.
Well, I don't know what
that new direction is.
Maybe 1, 0, 0.
What the heck?
I believe-- I hope I'm
right-- that 1, 0, 0
is not a combination here.
I say 1, 0, 0 goes off.
It's pretty short.
Here's a3.
Better a3 then
that loser 6, 3, 5.
1 0, 0 is a winner.
These three vectors--
So now a1, a2, and let me
add in a3, all three of them
span a-- what do they span?
What are all the
combinations of a1, a2, a3?
It's three dimensional?
It's the whole three
dimensional space.
They span all of 3D, the
whole three dimensional space.
They're a basis for the whole
three dimensional space.
They're independent.
So let me-- you see that
picture before I move it?
a1, a2, a3 are independent.
None of them is a
combination of the others.
They fill a three
dimensional space.
They're are a basis for that
three dimensional space.
And that space is,
in this example,
is the whole of our three.
So let me just write down on
the next blackboard what I mean.
Independent.
Independent.
So independent
columns of a matrix.
Independent columns of a matrix
A means the only solution to Av
equals 0 is v equals 0.
So if I have
independent columns,
then I haven't got
any null space.
If I have independent
columns, then the null space
of the matrix is
just the 0 vector.
So let me write down
that example again.
A was the matrix 2, 1,
5, 4, 2, 0, 1, 0, 0.
So I believe that matrix
has independent columns.
So its column space is the
full three dimensional space.
It's null space
only contains-- let
me put it, make that clear
that that's a vector.
And now I'm ready to write
down the idea of a basis.
So what is a basis
for the space?
A basis for a space, a subspace.
Independent vectors.
That's the key.
Independent vectors that
span the space, the subspace.
Whatever it is.
By the way, if the
column space is all
a three dimensional space, as it
is here, that's a subspace too.
It's the whole space,
but the whole space
counts as a subspace of itself.
And the 0 vector alone counts
as the smallest possible.
So if we're in three dimensions,
the idea of subspaces
has-- we have just the 0 vector.
Just one point.
That's a smallest.
We have the whole three
dimensional space.
That's the biggest.
And then we have all
the lines through 0.
Those are on the small side.
We have all the
planes through 0.
Those are a bit bigger.
And those dimensions
are 0, 1, 2, 3.
The possible dimensions
is told to us
by how many basis
vectors we need.
So let me look at that and
then come to dimension.
OK.
So independent means
that the only--
that no combination,
no other combination
of the vectors, no
combination of these vectors
gives the 0 vector except to
take 0 of that, 0 of that,
and 0 of that.
So those are a basis
for the column space
because they're independent
and their combinations
give the whole column space.
OK.
And now I wanted to say
something about dimensions.
OK.
Dimension.
It's a number.
It's the number of basis
vectors for the subspace.
Oh!
But you might say,
that the subspace
has other bases,
not just the one you
happen to think of first.
And I agree.
Many different bases.
For this example, all I need to
get a basis for, in this case,
for three dimensional space is I
need three independent vectors.
Any three.
But the point is, the
point about dimension
is that I need exactly three.
I can never get two vectors
that span all of our three.
And I can never get
four vectors that
are independent in our three.
If I have fewer than
the dimension number,
I don't have enough.
They don't span.
If I have too many, than the
dimension, they're dependent.
They won't be independent.
They can't be a basis.
Every basis has the same number.
And that number is the
dimension of the subspace.
All right, let's just take an
example, just with a picture.
I'll stay in three
dimensional space,
but my subspace will
just be a plane.
So here I'm in three
dimensional space.
Good.
Now I have my
subspace is a plane.
So it goes through the
origin, but it's only a plane.
So I'm expecting that I could
take a vector in the plane,
and I could take another
vector in the plane,
and they could be independent.
They are.
They're different directions.
I couldn't find a third
independent vector
in the plane.
Every basis for the plane--
So here every basis for this
plane contains two vectors.
Always two.
And that number two is
the dimension of a plane.
Well, I'm just saying the
plane there is two dimensional.
It's not the same as r2.
it's not the same.
That plane is a plane in r3.
It's not ordinary two
dimensional space.
But its dimension is two
because it takes any vector.
And if I didn't like
the looks of this one,
well, that's no problem.
Let me go that way.
That's just as good.
Those two vectors
are independent.
They span the plane.
They're a basis for the plane.
The plane is two dimensional.
That's the set of key ideas.
Independent.
Span.
Basis.
Basis is fundamental.
Basis is a bunch of vectors.
And dimension is
how many vectors.
OK.
Those are key ideas
in linear algebra.
And you'll see them come
into the big picture
of linear algebra.
Thank you.
