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