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PROFESSOR: So as long
as I'm introducing

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the idea of a vector
space, I better introduce

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the things that go with it.

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The idea of its dimension
and, all important, the idea

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of a basis for that space.

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That space could be all of
three dimensional space,

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the space we live in.

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In that, case the
dimension is three,

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but what's the meaning of
a basis-- a basis for three

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dimensional space.

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Or a basis for other spaces.

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OK, so I have to explain
independence, basis,

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and dimension.

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Dimension's easy if
you get the first two.

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OK, independence.

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Are those vectors independent?

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Well, if I draw them, in
three dimensional space,

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I can imagine 2, 1, 5
going in some direction.

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Let me draw it.

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How's that?

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2, 1, 5, whatever!

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Goes there.

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That's a1.

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OK.

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Now is a2 on the same line?

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If a2 is on the same line
then it would be dependent.

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The two vectors
would be dependent

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if they're on the same line.

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But this one is
not on that line.

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A 4, 2, 0.

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So it doesn't go up and all.

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It's somewhere in
this plane, 4, 2, 0.

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I'll say there.

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Whatever.

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a2.

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So those are independent.

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So their combinations
give me a space.

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The combinations of a1 and a2
give me a plane, a flat plane,

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in three dimensional space.

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That plane is, I would
say, they span the plane.

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a1 and a2 span a plane.

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And here's the key word: span.

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So there are two vectors.

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They're in three
dimensional space.

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And the plane they span
is all their combinations.

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That's what we're always doing:
taking all the combinations

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of these vectors.

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OK.

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So there-- and actually, a1 and
a2 are a basis for that pane.

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a1 and a2 are a
basis for that plane

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because their combinations
fill the plane.

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And also, they're independent.

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I need them both.

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If I threw away one, I would
only have one vector left,

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and it would only span a line.

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OK.

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Now let me bring in a third
vector in three dimensions.

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Well, what shall I take
for that third vector?

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Ha!

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Suppose I take a1 plus
a2 as my third vector.

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So 6, 3, 5.

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What about the vector 6, 3, 5?

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Well, what do I know?

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It's obviously special.

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It's a1 plus a2.

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It's in the same plane.

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So if I took a3 equal 6, 3,
5, that would be dependent.

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The three vectors would
be dependent with that a3.

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They would span the plane still.

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Their combinations would
still give the plane,

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but they wouldn't be
a basis for the plane.

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a1 and 12 and a3 together,
that's too much, too many

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vectors for a single plane.

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The vectors are dependent.

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And we don't-- a basis has
to be independent vectors.

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You have to need them all.

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We don't need all three here.

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So that's a dependent one.

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It can't go into a
basis with a1 and a2

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because the three
vectors are dependent.

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Now let me make a
difference choice.

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So that one's dead.

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That did not do it.

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All right.

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Let me take a3
equal to some other,

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not a combination of
these, but headed off

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in some new direction.

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Well, I don't know what
that new direction is.

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Maybe 1, 0, 0.

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What the heck?

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I believe-- I hope I'm
right-- that 1, 0, 0

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is not a combination here.

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I say 1, 0, 0 goes off.

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It's pretty short.

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Here's a3.

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Better a3 then
that loser 6, 3, 5.

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1 0, 0 is a winner.

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These three vectors--

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So now a1, a2, and let me
add in a3, all three of them

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span a-- what do they span?

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What are all the
combinations of a1, a2, a3?

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It's three dimensional?

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It's the whole three
dimensional space.

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They span all of 3D, the
whole three dimensional space.

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They're a basis for the whole
three dimensional space.

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They're independent.

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So let me-- you see that
picture before I move it?

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a1, a2, a3 are independent.

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None of them is a
combination of the others.

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They fill a three
dimensional space.

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They're are a basis for that
three dimensional space.

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And that space is,
in this example,

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is the whole of our three.

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So let me just write down on
the next blackboard what I mean.

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Independent.

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Independent.

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So independent
columns of a matrix.

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Independent columns of a matrix
A means the only solution to Av

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equals 0 is v equals 0.

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So if I have
independent columns,

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then I haven't got
any null space.

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If I have independent
columns, then the null space

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of the matrix is
just the 0 vector.

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So let me write down
that example again.

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A was the matrix 2, 1,
5, 4, 2, 0, 1, 0, 0.

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So I believe that matrix
has independent columns.

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So its column space is the
full three dimensional space.

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It's null space
only contains-- let

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me put it, make that clear
that that's a vector.

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And now I'm ready to write
down the idea of a basis.

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So what is a basis
for the space?

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A basis for a space, a subspace.

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Independent vectors.

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That's the key.

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Independent vectors that
span the space, the subspace.

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Whatever it is.

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By the way, if the
column space is all

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a three dimensional space, as it
is here, that's a subspace too.

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It's the whole space,
but the whole space

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counts as a subspace of itself.

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And the 0 vector alone counts
as the smallest possible.

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So if we're in three dimensions,
the idea of subspaces

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has-- we have just the 0 vector.

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Just one point.

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That's a smallest.

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We have the whole three
dimensional space.

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That's the biggest.

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And then we have all
the lines through 0.

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Those are on the small side.

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We have all the
planes through 0.

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Those are a bit bigger.

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And those dimensions
are 0, 1, 2, 3.

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The possible dimensions
is told to us

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by how many basis
vectors we need.

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So let me look at that and
then come to dimension.

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OK.

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So independent means
that the only--

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that no combination,
no other combination

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of the vectors, no
combination of these vectors

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gives the 0 vector except to
take 0 of that, 0 of that,

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and 0 of that.

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So those are a basis
for the column space

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because they're independent
and their combinations

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give the whole column space.

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OK.

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And now I wanted to say
something about dimensions.

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OK.

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Dimension.

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It's a number.

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It's the number of basis
vectors for the subspace.

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Oh!

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But you might say,
that the subspace

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has other bases,
not just the one you

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happen to think of first.

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And I agree.

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Many different bases.

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For this example, all I need to
get a basis for, in this case,

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for three dimensional space is I
need three independent vectors.

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Any three.

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But the point is, the
point about dimension

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is that I need exactly three.

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I can never get two vectors
that span all of our three.

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And I can never get
four vectors that

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are independent in our three.

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If I have fewer than
the dimension number,

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I don't have enough.

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They don't span.

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If I have too many, than the
dimension, they're dependent.

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They won't be independent.

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They can't be a basis.

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Every basis has the same number.

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And that number is the
dimension of the subspace.

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All right, let's just take an
example, just with a picture.

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I'll stay in three
dimensional space,

195
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but my subspace will
just be a plane.

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So here I'm in three
dimensional space.

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Good.

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Now I have my
subspace is a plane.

199
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So it goes through the
origin, but it's only a plane.

200
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So I'm expecting that I could
take a vector in the plane,

201
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and I could take another
vector in the plane,

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and they could be independent.

203
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They are.

204
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They're different directions.

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I couldn't find a third
independent vector

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in the plane.

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Every basis for the plane--

208
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So here every basis for this
plane contains two vectors.

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Always two.

210
00:12:06,930 --> 00:12:11,160
And that number two is
the dimension of a plane.

211
00:12:11,160 --> 00:12:15,080
Well, I'm just saying the
plane there is two dimensional.

212
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It's not the same as r2.

213
00:12:18,380 --> 00:12:19,510
it's not the same.

214
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That plane is a plane in r3.

215
00:12:22,910 --> 00:12:26,090
It's not ordinary two
dimensional space.

216
00:12:26,090 --> 00:12:30,420
But its dimension is two
because it takes any vector.

217
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And if I didn't like
the looks of this one,

218
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well, that's no problem.

219
00:12:35,450 --> 00:12:38,260
Let me go that way.

220
00:12:38,260 --> 00:12:39,910
That's just as good.

221
00:12:39,910 --> 00:12:42,250
Those two vectors
are independent.

222
00:12:42,250 --> 00:12:44,210
They span the plane.

223
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They're a basis for the plane.

224
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The plane is two dimensional.

225
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That's the set of key ideas.

226
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Independent.

227
00:12:51,470 --> 00:12:53,810
Span.

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Basis.

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00:12:54,470 --> 00:12:56,560
Basis is fundamental.

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00:12:56,560 --> 00:12:59,040
Basis is a bunch of vectors.

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00:12:59,040 --> 00:13:02,381
And dimension is
how many vectors.

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00:13:02,381 --> 00:13:02,880
OK.

233
00:13:02,880 --> 00:13:05,350
Those are key ideas
in linear algebra.

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00:13:05,350 --> 00:13:11,910
And you'll see them come
into the big picture

235
00:13:11,910 --> 00:13:13,400
of linear algebra.

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00:13:13,400 --> 00:13:15,170
Thank you.