MARTINA BALAGOVIC: Hi.
Welcome back.
Today's problem is about solving homogeneous linear systems,
A*x equals 0, but it's also an introduction to the next
lecture and next recitation section,
which are going to be about solving non-homogeneous linear
systems, A*x equals b.
The problem is fill the blanks type.
And it says the set S of all points with coordinates x, y,
and z, such that x minus 5y plus 2z equals 9 is a blank in R^3.
It is in a certain relation to the other blank S_0
of all the points with coordinates x, y,
and z that satisfy the following linear equation,
x minus 5y plus 2z equals 0.
After we solve this, we have the second part
of the problem, which says all points of x have
a specific form, x, y, z equals blank, 0,
0, plus some parameter times blank, 1,
0 plus some other parameter times blank, 0, 1.
And we need to fill out all six blanks.
Now you should pause the video, fill in the blanks,
and then come back and see some pretty pictures
that I prepared for you.
And we're back.
So you probably picked this up in lectures already.
If you have a three-dimensional space with three degrees
of freedom, and put in one constraint, so put
in one equation, you get something
that has two degrees of freedom, something
that's two-dimensional.
If this equation is linear, rather than quadratic
or cubic or exponential, this something
is something two-dimensional and flat.
Something two-dimensional and flat in R^3 is also called
a plane, or a two-plane.
Similarly, S_0 is also a plane.
Now, what's the relation between S and S_0
if they're given by these two equations?
Well first let's look at the general positions in which two
planes in R^3 can be.
First one is that they're intersecting along a line.
What's going to happen here is that all points on this plane
are points whose coordinates satisfy
the equation of this plane.
The points in this plane are points
whose coordinates satisfy the equation of this plane.
And the points on the line are points whose coordinates
satisfy the system of this equation and this equation.
The other position in which two planes can be
is that they're not intersecting at all, that they're parallel.
So let's start by trying to find this line here.
The equation of one plane is x minus 5y plus 2z equal 9.
The equation of the other one is x minus 5y plus 2z equals 0.
Now you can just look at it and see
how many solutions it's supposed to have,
or you can try doing elimination,
and after one step of elimination
get 0 equals 9, which never happens.
There cannot exist numbers x, y, and z such that this
combination of them produces 0, and the same combination
of them produces 9 at the same time.
So this red line here doesn't exist,
and the situation of these two planes S and S_0 is this one,
they're parallel.
So let's add the word parallel in here.
And let's move on to the other half of the problem.
The other half said all points of S have this specific form.
Now let me call this point here P_0.
If all points of S have this form,
we can plug in any parameter c_1 and c_2 here
and we're going to get a point of the plane.
So in particular, we can plug in c_1 and c_2 equal to 0.
What we get then is that the point (x, y, z)
equals P_0 is a point of the plane S. So P_0 is in S.
What do we know about the point P_0?
Well the fact that it's in S means
that its coordinates, x minus 5y plus 2z equal 9.
That's the equation of S. But we also
know that y and z are equal to 0 and 0.
Solving this system we get that the x-coordinate of this point
P_0 is 9, and we can just add 9 here.
So we just have two blanks left to fill.
Before we'll fill them, let me show you
a picture that I drew here.
So we have these two planes, S_0 and S, which are parallel.
They're given by these equations.
And the plane S_0 has a point 0 in it,
because the equation is x minus 5y plus 2z equals 0,
so it satisfied by 0, 0, 0.
The plane S has this point P_0 in it, which is (9, 0, 0)--
we just figured this out.
And there's this vector connecting one plane
to the other.
Now, since those two planes are parallel
and there's this vector going between them, what we can see
is that a good way to get any point in S
is to go to any point in S_0 and go up by this vector.
Now let me write this down.
What I just said is that any point in S is of the form--
use this vector to go up-- plus any point in S_0.
And if we compare this to this expression here,
we also get P_0 plus this linear combination.
So this here has to be a point in S_0.
Now we're left with a question of how to parameterize
all points in S_0.
What are all the points in S_0, and what does this problem
have to do with solving homogeneous linear equations?
Well, let me write this equation of S_0
in a slightly different way.
Let me write it as 1, minus 5, 2, x, y, z, equals 0.
And let me think of this as a matrix of the system.
It's a very tiny matrix, but it's a matrix.
And think of it as a matrix dot a vector equals
0, and trying to find all solutions of the system.
Well let's do row reductions here.
It's already as upper triangular as these tiny matrices get.
This is a pivot.
So we have a pivot variable x.
These are free variables, y and z.
And if you remember how to solve these systems,
for each free variable we get one particular solution.
So we get one particular solution when we plug in y is 1
and all the other free variables are 0.
Plugging it in here, we just get that in that case,
x-- so we get x minus 5 times 1 plus 2 times 0 equals 0.
So x is equal to 5.
And the other solution is for setting
all free variables equal to 0, except z
which we set equal to 1.
And then we get x minus 5 times 0 plus 2 times 1 equals 0.
So we get that in this case, x equals minus 2.
And any solution of this system is going to be of the form
some constant times this plus some other constant times this.
And if we walk back to our original problem here,
we see that these parameters, these numbers here,
have been set up exactly so that we can just take these numbers
and just copy them over, 5 and minus 2.
And this is the general form of any point of the plane S.
It's go up this vector, and then add a point in S_0,
in the parallel plane that passes through the origin.
This finishes our problem.
But what I would encourage you to do now
is to go on to the next lecture, watch the next recitation
video, and then come back here and think
about what is it that we really did here
on this half of the board.
Thank you.