PROFESSOR: Hi, welcome back to recitation.
In the lecture, we've learned very important concepts--
linear space and linear subspace.
Well, as you can imagine, if we call something a space,
we're putting a lot of things, a lot of objects into one set.
But for linear space, we want to put them in according to a
particular manner.
So can you recall what are the conditions for a set to be a
linear space?
You take any two elements from that set,
take the sum of them.
You want the sum to be still in the same set.
That's the first condition.
Second, you take any multiple of any element from that set,
the result will still be in that set.
That's the second condition.
And if within a linear space you can find a subset which
also satisfies the two conditions, that will give you
a subspace.
Today, we're going to look at this example to review these
two important properties of linear space and subspace.
I have two vectors, x1 and x2.
Both of them are vectors in R3.
So as you can see, I've drawn them here.
This is x1, and this is x2.
So first, we want to find the subspace generated by x1.
I call it v1.
Let me say a word about this generated by.
So what do I mean by a subspace generated by x1?
I'm looking for the smallest subspace that contains x1, as
small as you can get.
Similarly, I want to find out the subspace generated by x2,
call it v2.
Then we want to say something about the
intersection of v1 and v2.
That's the first question.
And the second question, we want to put vector x1 and x2
together and look at the subspace generated by x1 and
x2 at the same time.
So I call it v3.
And a good question to be asked here is what is the
relation of v3 to v1 union v2.
Do you think they're equal?
Within the second question, I would also like you to find a
subspace, call it s of v3, such that neither x1
nor x2 is in s.
And the last question is I'd like you to say something
about the intersection of v3 with the xy plane.
So notice that, of course, xy plane is also
a subspace of R3.
So again, I'm looking at the intersection of two subspaces.
All right, why don't you hit the pause now, and try to
solve these three problems on your own.
And I'd like you to identify your answers in this picture
whenever you can.
I'll come back later and continue working with you.
OK, how did your drawing go?
Let's look at this together.
First, we want to find subspace generated by x1.
So here is x1.
Let's keep in mind the two conditions that a subspace has
to satisfy.
Well, if I want to obtain the subspace, at least I have to
be able to take any multiple of x1, right?
So that means at least I have to include the straight line
that contains x1.
So I'm going to try to draw the straight line here.
So I'm just simply going to extend this x1 along to both
directions.
So x1, I will try very hard to make it straight.
But to be honest, it's really hard for me to draw a straight
line from the board.
Is that straight?
Seems fine.
All right, so this entire line contains x1.
So at least this line has to be in v1.
Is there anything else beyond this line?
Now let's turn to the second condition.
The second condition says that we have to be able to take any
two elements, take the sum, and the sum will
remain in that set.
Does it work for this line?
You take any two vectors on this line or you can say any
two points on this line, and then you take to sum of them.
Of course, you still get something on this line.
You won't be able to escape from it.
Which means this line is a perfect set that satisfies the
two conditions.
So this line simply gives me v1.
That is the smallest subspace that contains x1.
So in other words, the subspace
generated by x1 is v1.
So similarly, let's look at x2.
What is the subspace generated by x2?
Again you get the entire straight line
that contains x2.
So I'm going to extend x2 in both directions.
I hope it's straight.
Not too bad.
That will give me v2.
All right, v1, v2.
Both of them are subspaces of R3.
Now let's look at the intersection of v1 and v2.
So we know that both of them are straight lines.
And clearly they're not parallel, because x1 and x2
are not parallel.
So what is the intersection of v1 and v2?
The intersection of v1 and v2 is the only point at which
they cross.
And where's that point?
It's here, right at the origin.
Because both of them pass the origin.
I'm going to use o to denote.
That's the intersection of v1 and v2.
This is a set with only one element.
What can you say about this set?
I claim this is also a subspace of R3.
By saying space, we usually mean a
lot of objects together.
But look at this set.
This set fits perfectly into the conditions of being a
linear space.
You take any multiple of 0, again you get 0--
0 plus 0, you get 0.
So that's a perfectly fine subspace of R3.
All right, so what we have got here is I take the
intersection of v1 and v2, and the result, again, becomes a
subspace, which only contains the origin.
That completes the first question.
Let's look at the second one.
In the second question, I want to put x1 and x2 together, and
look at the subspace generated by x1 and x2.
And I would also like you to say something about the
relation between v1 union v2 to the subspace
generated by x1 x2.
Let's try to answer the second question first. Is there a
chance that v1 union v2 equal to v3?
OK, so what is v1 union v2?
That's clearly just two lines, right?
This line union this line.
Is there a chance that this union will be a subspace?
Let's check the two conditions.
First, you take any multiple of the elements in this union.
It's either on this line or on this line.
Seems that the multiple is still going to
stay inside the union.
So the first condition is actually satisfied.
What about the second one?
The second one says that I have to be able to take any
sum, the sum of any two elements,
from these two lines.
Let's just try a simple sum, x1 plus x2.
So what is x1 plus x2?
You just sum up each coordinate.
That will give you 2, 5, 3.
In this picture, can I draw it in this picture?
It's going to be somewhere here.
That's x1 plus x3.
Did you notice something?
You clearly have got out of this union.
So this sum is not inside v1 union v2.
Which means v1 union v2 is not a subspace.
Then it's impossible that this union will equal to v3.
So the answer to the second question is no, v3 is not
equal to v1 union v2.
Now let's identify v3.
Well, as you can see, since we have seen from this argument
that we have to be able to include this
diagonal vector here.
But in fact, as you can see since we can take any elements
from these two lines, we're actually including every
vector on the plane spanned by v1 and v2.
So in other words, I'm actually looking at this huge
plane that is spanned by v1 and v2.
That will give me v3.
So that's reasonable.
I'm looking at the subspace generated by two lines in R3.
And that two line will be able to span a plane in R3.
Now for the last part of question two, I want to find a
subspace s of v3.
So I want to find a subspace of this plane such
that x1 is not s.
x2 is not in s either.
So I want to stay away from this line and this line.
Can you find such a subspace of v3?
It's right here.
Because if you look at this vector, x1 plus x2-- sorry,
that should be x2.
If you look at this vector, and if you look at the
subspace generated by this vector, again you know it's
going to be a line.
And this line is right here.
This line forms a perfect subspace of v3.
But neither x1 nor x2 is inside this subspace.
So let's just make it S. Of course, the
choice is not unique.
You can take twice x1 plus x2 or x1 plus twice x2.
OK, we have completed the second problem.
The last problems ask us to find the intersection of v3
with xy plane.
Well, just think about that.
We've identified v3 as this plane spanned
by line v1 and v2.
And xy is also a plane.
So we're talking about the intersection of
two planes in R3.
What would that be?
So you have two planes intersect.
The intersection will be a straight line again, right?
So let's locate that straight line.
We want to find something that is inside of v3 and xy plane
at the same time.
What can you say about the points in xy plane?
The z-coordinate has to be 0.
And at the same time, we know that at least x1
and x2 are in v3.
So did you notice that?
x2 is a vector that lies in v3.
But at the same time, the z-coordinate of x2 is 0.
That is what we're looking for.
So the intersection of v3 with xy plane will simply be the
line that contains x2.
And here we've identified that as v2.
That's it.
This is a subspace of R3.
And this is the subspace of R3.
The intersection is again a subspace of R3.
I hope you've learned a way to somehow visualize this linear
space and subspace through this exercise.
Thank you for watching, and I'll see you next time.