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PROFESSOR: Hi, welcome
back to recitation.
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In the lecture, we've learned
very important concepts--
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linear space and
linear subspace.
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Well, as you can imagine, if
we call something a space,
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we're putting a lot of things,
a lot of objects into one set.
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But for linear space, we
want to put them in according
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to a particular manner.
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So can you recall what are
the conditions for a set
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to be a linear space?
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You take any two elements from
that set, take the sum of them.
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You want the sum to be
still in the same set.
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That's the first condition.
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Second, you take any multiple
of any element from that set,
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the result will
still be in that set.
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That's the second condition.
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And if, within a
linear space, you
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can find a subset which also
satisfies the two conditions,
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that will give you a subspace.
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Today, we're going to
look at this example
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to review these two important
properties of linear space
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and subspace.
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I have two vectors, x_1 and x_2.
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Both of them are vectors in R^3.
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So, as you can see,
I've drawn them here.
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This is x_1, and this is x_2.
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So first, we want to find the
subspace generated by x_1.
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I call it V_1.
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Let me say a word about
this "generated by."
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So what do I mean by a
subspace generated by x_1?
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I'm looking for the smallest
subspace that contains x_1, as
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small as you can get.
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Similarly, I want to find out
the subspace generated by x_2,
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call it V_2.
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Then we want to say something
about the intersection of V_1
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and V_2.
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That's the first question.
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And the second question, we
want to put vector x_1 and x_2
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together and look at the
subspace generated by x_1
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and x_2 at the same time.
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So I call it V_3.
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And a good question
to be asked here
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is: what is the relation
of v3 to V_1 union V_2?
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Do you think they're equal?
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Within the second
question, I would also
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like you to find a subspace,
call it S, of V_3 such
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that neither x_1
nor x_2 is in s.
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And the last
question is I'd like
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you to say something about
the intersection of V_3
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with the xy-plane.
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So notice that, of course,
xy-plane is also a subspace
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of R^3.
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So again, I'm looking at the
intersection of two subspaces.
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All right, why don't
you hit the pause now,
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and try to solve these
three problems on your own.
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And I'd like you to identify
your answers in this picture
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whenever you can.
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I'll come back later and
continue working with you.
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OK, how did your drawing go?
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Let's look at this together.
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First, we want to find
subspace generated by x_1.
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So here is x_1.
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Let's keep in mind
the two conditions
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that a subspace has to satisfy.
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Well, if I want to
obtain the subspace,
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at least I have to be able
to take any multiple of x_1,
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right?
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So that means at least I have to
include the straight line that
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contains x_1.
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So I'm going to try to draw
the straight line here.
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So I'm just simply going
to extend this x1 along
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to both directions.
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So x_1, I will try very
hard to make it straight.
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But to be honest, it's
really hard for me
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to draw straight
lines on the board.
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Is that straight?
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Seems fine.
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All right, so this
entire line contains x_1.
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So at least this line
has to be in V_1.
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Is there anything
else beyond this line?
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Now let's turn to
the second condition.
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The second condition
says that we
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have to be able to take any
two elements, take the sum,
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and the sum will
remain in that set.
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Does it work for this line?
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You take any two
vectors on this line,
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or you can say any two
points on this line,
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and then you take
to sum of them.
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Of course, you still get
something on this line.
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You won't be able
to escape from it.
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Which means this line
is a perfect set that
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satisfies the two conditions.
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So this line simply
gives me V_1.
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That is the smallest
subspace that contains x_1.
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So in other words, the subspace
generated by x_1 is V_1.
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So similarly, let's look at x_2.
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What is the subspace
generated by x_2?
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Again, you get the
entire straight line
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that contains x_2.
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So I'm going to extend
x_2 in both directions.
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I hope it's straight.
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Not too bad.
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That will give me V_2.
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All right, V_1, V_2.
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Both of them are
subspaces of R^3.
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Now let's look at the
intersection of V_1 and V_2.
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So we know that both of
them are straight lines.
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And clearly they're
not parallel,
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because x_1 and x_2
are not parallel.
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So what is the intersection
of V_1 and V_2?
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The intersection of V_1
and V_2 is the only point
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at which they cross.
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And where's that point?
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It's here, right at the origin.
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Because both of them
pass the origin.
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I'm going to use O to denote.
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That's the intersection
of V_1 and V_2.
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This is a set with
only one element.
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What can you say about this set?
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I claim this is also
a subspace of R^3.
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By saying space, we usually
mean a lot of objects together.
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But look at this set.
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This set fits perfectly
into the conditions
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of being a linear space.
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You take any multiple of 0,
again you get 0-- 0 plus 0,
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you get 0.
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So that's a perfectly
fine subspace of R^3.
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All right, so what
we have got here is:
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I take the intersection
of V_1 and V_2,
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and the result, again,
becomes a subspace,
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which only contains the origin.
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That completes the
first question.
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Let's look at the second one.
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In the second question, I want
to put x_1 and x_2 together,
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and look at the subspace
generated by x_1 and x_2.
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And I would also like
you to say something
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about the relation
between V_1 union V_2
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to the subspace
generated by x_1, x_2.
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Let's try to answer the
second question first.
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Is there a chance that
V_1 union V_2 equal to v3?
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OK, so what is V_1 union V_2?
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That's clearly just
two lines, right?
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This line union this line.
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Is there a chance that this
union will be a subspace?
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Let's check the two conditions.
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First, you take any multiple
of the elements in this union.
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It's either on this
line or on this line.
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Seems that the multiple is still
going to stay inside the union.
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So the first condition
is actually satisfied.
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What about the second one?
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The second one says
that I have to be
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able to take any sum, the
sum of any two elements,
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from these two lines.
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Let's just try a simple
sum, x_1 plus x_2.
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So what is x_1 plus x_2?
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You just sum up each coordinate.
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That will give you [2, 5, 3].
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In this picture-- can I
draw it in this picture?
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It's going to be somewhere here.
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That's x_1 plus x_3.
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Did you notice something?
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You clearly have got
out of this union.
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So this sum is not
inside V_1 union V_2.
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Which means V_1 union
V-2 is not a subspace.
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Then it's impossible that
this union will equal to V_3.
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So the answer to
the second question
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is no, V_3 is not
equal to V_1 union V_2.
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Now let's identify V_3.
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Well, as you can see, since we
have seen from this argument
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that we have to
be able to include
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this diagonal vector here.
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But in fact, as you can see
since we can take any elements
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from these two
lines, we're actually
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including every vector on the
plane spanned by V_1 and V_2.
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So in other words, I'm actually
looking at this huge plane
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that is spanned by V_1 and V_2.
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That will give me V_3.
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So that's reasonable.
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I'm looking at the subspace
generated by two lines in R^3.
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And that two line will be
able to span a plane in R^3.
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Now for the last
part of question two,
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I want to find a
subspace S of V_3--
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so I want to find a
subspace of this plane--
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such that x_1 is not S,
x_2 is not in S either.
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So I want to stay away from
this line and this line.
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Can you find such
a subspace of V_3?
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It's right here.
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Because if you look at this
vector, x_1 plus x_2-- sorry,
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that should be x_2.
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If you look at this
vector, and if you
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look at the subspace
generated by this vector,
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again you know it's
going to be a line.
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And this line is right here.
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This line forms a
perfect subspace of V_3.
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But neither x_1 nor x_2
is inside this subspace.
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So let's just make it S. Of
course, the choice is not
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unique.
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You can take twice x_1 plus
x_2 or x_1 plus twice x_2.
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OK, we have completed
the second problem.
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The last problems ask us to
find the intersection of V_3
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with xy-plane.
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Well, just think about that.
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We've identified
V_3 as this plane
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spanned by line V_1 and V_2.
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And xy is also a plane.
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So we're talking about the
intersection of two planes
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in R^3.
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What would that be?
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So you have two
planes intersect.
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The intersection will be a
straight line again, right?
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So let's locate
that straight line.
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We want to find something that
is inside V_3 and xy-plane
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at the same time.
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What can you say about
the points in xy-plane?
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The z-coordinate has to be 0.
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And at the same time, we know
that at least x_1 and x_2
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are in V_3.
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So did you notice that?
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x_2 is a vector
that lies in V_3.
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But at the same time, the
z-coordinate of x_2 is 0.
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That is what we're looking for.
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So the intersection
of V_3 with xy-plane
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will simply be the
line that contains x_2.
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And here we've
identified that as V_2.
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That's it.
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This is a subspace of R^3.
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And this is the subspace of R^3.
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The intersection is
again a subspace of R^3.
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I hope you've learned a
way to somehow visualize
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this linear space and subspace
through this exercise.
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Thank you for watching,
and I'll see you next time.