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Let's say I have some subspace
V, that is a subset of Rn. And let's say that we also
have its orthogonal complement, we write
that as V perp. That'll also be a
subset of Rn. A couple of videos ago, it might
have even been the last video if I remember properly, we
learned that the dimension of V, plus the dimension of the
orthogonal complement of V, which is also another
subspace, is going to be equal to n. Remember dimension is just the
number of linearly independent vectors you need to have
a basis for V. And the dimension here is the
number of linearly independent vectors you need to have a
basis for the orthogonal complement of V. Now given this, let's see if we
can come up with some other interesting ways in which these
two subspaces relate to each other. Or how they might relate to
all of the vectors in Rn. So the first question is, do
these two subspaces have anything in common? Are there any vectors that are
in common with the two. And to test whether there is,
let's just assume there is, and see what the properties of
that vector would have to be. Let's assume right here that I
have some vector x that is a member of my subspace V. Let's also assume that x is
a member of the orthogonal complement of V. Now what does this second
statement mean? Membership in the orthogonal
complement means that x dot v, for any v that is a member of
our subspace, is going to be equal to 0. Let me write it this
way actually. x dot v is equal to 0
for any v that is a member of our subspace. That's what it means to be a
member of V's orthogonal complement. Now we assume that x is
also a member of V. So that means that we can
stick x here as well. For any member of V. x is also a member of V. So that implies that x
dot x is equal to 0. Another way to write that is
that the length of x squared is equal to 0. Or the length of x
is equal to 0. And that's only true
for one vector. You can even try it out
with the different constituents of x. The only vector that that's
true for is the 0 vector. So x has to be equal
to the 0 vector. That's the only vector in Rn
that when you dot it with itself you get 0, or whose
the square of its length is equal to 0. And we've shown that many,
many, many videos ago. What this tells us is at that
the intersection between V and V complement-- this kind of
upside down U just means intersection, it just means
where do these two sets overlap-- the only place that
these overlap is with a subset of the 0 vector. So if I were to draw all
of Rn like this. Let's say that this is Rn. And let's say I draw
the subspace V. And let's say I draw the
orthogonal complement to V. It's all of these vectors
right here. This is the orthogonal
complement to V right there. So this is V perp. These are all of the vectors
that when I dot it with any vector here, I'm going
to get equal to 0. So this is V perp. The intersection, their overlap,
the only vector that is a member of both
is the 0 vector. That's their only
intersection. So that's fair enough. The only vector that's a member
of a subspace and its orthogonal complement
is the 0 vector. Nothing too profound there. Let's see if we can come up with
some other interesting relations between the subspace
and its orthogonal complement. Maybe some arbitrary
vectors in Rn. So let's just write down--
well we know that the dimension of our subspace
V is equal to k. If its equal to k, we know that
its dimension plus its orthogonal complement has to be
equal to n, because we're dealing in Rn. And we also know that the
orthogonal complement of V is a subset of Rn, I drew
it right here. The dimension of V
is equal to k. That's a k right there. And what's the dimension of the
orthogonal complement of V going to be? Well, when you add them
together-- I wrote that up here-- they have to equal n. So this guy's going to
have to be n minus k. If you have k here. This guy's dimension is k, this
guy's dimension right here if, it's n minus k, that
when you add these two up, k plus n minus k is going
to be equal to n. So this guy will have a
dimension of n minus k. Now what does dimension mean? It means that that's the number
of linearly independent vectors you need to
form a basis. I have k vectors as
a basis for V. I have v1, v2, all
the way to vk. And this is a basis for V, which
just means they're all linearly independent. And they span V. Any member of V right here can
be represented as a linear combination of these vectors. Now the dimension of the
orthogonal complement of V is n minus k. So we could have n
minus k vectors. Let's call them w1, w2, all
the way to wn minus k. We have n minus k of
these characters. And this set is a basis
for the orthogonal complement of V. So any vector in here can be
represented as a linear combination of these
guys right here. And all of these guys are
linearly independent. So you don't have any redundant
vectors there. Now let's explore. And I'll tell you where
I'm trying to go. I'm trying to see if I combine
these two sets, whether I get a basis for all of Rn. That's what I'm trying
to understand. Let's just say that for some
constants c1 times v1 plus c2 times v2 plus all the way to
ck times vk plus-- for the constants on these guys I'll use
d-- plus d1 times w1 plus d2 times w2, all the way to
plus dn minus k times the basis vector wn minus k. Let's say that I'm curious about
setting this sum equal to 0, equaling the 0 vector
for some scalers. The scalers are these
c's and these d's. And we know that there's at
least one solution set of scalers for which
this is true. We could multiply all of these
constants-- c1, c2, ck, d1, d2, all the way to dn minus k. They could all be 0. Or there might be more
than one solution. In fact, if the only solution is
that all of these constants have to be equal to 0, then we
know that all of these vectors are linearly independent with
respect to each other. And if they're all linearly
independent with respect to each other, then we know that
they can be a basis for Rn. But we don't know that yet. We don't know that the only
solution to this is all of the constants being equal to 0. So let's see if we can
experiment with this a little bit. If we take this equation, which
I just wrote down, we know that one solution is all of
the constants, the c's and d's equalling to 0, but
we don't know that that's the only one. Let's just subtract all of the
w vectors from both sides of this equation. So what are we going to get? We're going to get c1, v1
plus c2, v2 all the way to plus ck, vk. And we're going to subtract
this from both sides of the equation. It's going to be equal
to the 0 vector. Which is really just 0, I don't
even have to write it down, but maybe I'll write
it down there just so you understand. I'm just taking this equation,
I'm subtracting these guys from both sides. So 0 vector minus d1, w1 plus
d2, w2 plus all the way to dn minus k, wn minus k. All I did is I subtract these
terms right here from both sides of this equation. I don't even have to write
this is 0 here, that's a bit redundant. So what I have here is I have
some combination of the basis vector of V. So if I look at this, this is
some linear combination of the basis vectors in V. If I call this a vector-- let
me call this some vector x. Let's say x is equal to
c1, v1 plus c2, v2 all the way to ck, vk. We know that it's a linear
combination of our basis vectors of V, so x
is a member of V. By definition, any linear
combination of the basis vectors for a subspace
is going to be a member of that subspace. Well, similarly, what do we have
on the right-hand side of this equation? On the right-hand side of this
equation, I have some linear combination of the basis vectors
of V complement You could put just put a minus all
along that, but that won't change the fact that this is
some linear combination of V complement's basis vectors. So this vector over here is
going to be a member of-- we could also call this x. So x is equal to this, but its
also going to be equal to this, and since it can be
represented as a linear combination of the orthogonal
complement of V's basis vector, or V perp's basis
vector, we know that this also has to be a member of V perp. Let me just review this,
because it can be a little bit confusing. I just set up this equation
right here. We know that there's at least
one solution-- all of the constants equalling 0. Anyone could do this. Now I subtracted all of the
yellow terms from both sides, and I got this equality. The left=hand side of this
equality is linear combinations of the basis
vectors of V. So any linear combinations of
the basis vectors of V is going to be a member of V. That's the definition
of basis vectors. So if I set x equalling to this
letf-hand side, I can say that x is a member of V. Well, if x is equal to the
left-hand side, it's also equal to the right-hand side. The right-hand side is some
linear combination of V perps, or the orthogonal complement
of V's basis vectors. Which tells us that x is also
a member of V perp. Well, what does that mean? That means that x must
be equal to 0. I just showed you at the
beginning of the video, the only vector that's a member of
a subspace and its complement is the 0 vector. So we know that because these
are orthogonal complements, we know that x must
be equal to 0. So just to reiterate, we know
0's has to equal both of these sides of the equation. And these are the same
constants that we had to begin with. But what do we know about
these two sets? We know that the 0 vector
has to be equal to this. That's the only vector in Rn
that's a member both of V and of the orthogonal
complement of V. Now, this is a 0 vector and we
have this linear combination of V's being set equaling
to the 0 vector. What do we know about
these constants? What does c1, c2, all the
way to ck have to be? We know that v1 through
vk is a basis for V. That tells us that they span V
and that they are linearly independent. Linear independence by
definition means that the only solution to this equation right
here is that all of the constants have to be 0. So linear independence tells
us that c1, c2, all the way through ck must be 0. All of these guys right
here are 0. Which is the same as
all of these guys. All of these guys must be 0. Now let's look at
the right-hand side of this equation. We could put the minus
all the way, but the same argument holds. This linear combination
of V perp's basis vectors is equal to 0. The only solution to this--
because each of these w1's, w2's, and wn minus k's are
linearly independent-- being equal to 0 is all of
the constants have to be equal to 0. That falls out of linear
independence. If this negative is confusing
you a bit, if it makes it look different than that, you could
just multiply this negative out and say minus d1 would have
to be equal to 0, minus d2 would have to be 0,
minus d and minus k would have to be 0. But it's the exact
same argument. Linear independence, which falls
out of the fact that this is a basis set, implies
that the only solution to this being equal to 0 is each
of the constants being equal to 0. Well, that means that d1,
d2, all the way to dn minus k must be 0. Let's go back to what
I wrote up here. This was the original equation
that we were experimenting with. Just manipulating this
equation a bit and understanding that the only
intersection between V and V perp is a 0 vector. And that you only have linear
independence if the only solution to these vectors
equalling 0 is all of their constants equalling 0. Then we know that all of these
terms, c1 through ck, d1 through dn minus k, they all
have to be equal to 0. That's the only solution to this
larger equation that I wrote up here. Well, the only solution to this
large equation that I wrote up here is that all of the
constants are equal to 0. That implies that if I were it
take the set right here of v1, v2, all the way to vk, and I
were to augment that with the basis vectors of V perp, which
are w1, w2, all the way to wn minus k, that this is a linearly
independent set. And I know that because the only
solution to this equation is each of these constants
having to be equal to 0. That's what linear independence
means. This implies this. Linear independence
implies that. We used the fact that linear
independence implies that all of these equal 0 to get the fact
that c1 all the way to ck was equal to 0. And then we use it again when
we set this thing also being equal to the 0 vector. We knew that all of the d's
had to be equal to 0. I don't know if you remember,
the 0 vector came out from the fact that that was the
only vector that is a member of both sets. I know I'm being a little bit
repetitive, but I really want you to understand that this
proof isn't some type of circular proof. That we just wrote this
equation, we wondered about what the solution set is to it,
we rearranged it, we said hey both sides of this equation
are members of both V and V perp. The only vector that's
a member of both is the 0 vector. So both of these sides
of the equation have to be equal to 0. The only solution to that is all
of these constants being equal to 0, because
each of these are linearly independent sets. So therefore all of these
constants have to be equal to 0. And then this augmented set,
where if you combined all of the basis vectors, that is
going to be linearly independent. Now many, many, many, many, many
videos ago, we learned that if we have some subspace
with dimension n, and we have n linearly independent vectors
that are members of your subspace, then those n linearly
independent vectors, or the set of your n vectors,
is a basis for the subspace. Now Rn is a subspace
of itself. Rn is an n dimensional
subspace. We could write the dimension
of Rn is equal to n. Now we have n linearly
independent vectors in Rn. So that tells us that these
guys right here are a basis for Rn. We have n linearly independent
vectors. We have n minus k that are
coming from V perp. We have n that are coming
from V from their basis for those subspace. So now we have a total
of n vectors. They're linearly independent. They're all members of Rn. So they are a basis for Rn. Which tells us that any vector
in Rn can be represented by linear combinations of these
guys, which is fascinating. So this is a basis for Rn. So that tells us that we can
take any vector-- let's say a is a member of Rn,
some vector. That means since this is a basis
for Rn that a can be represented to some linear
combination of all of these guys. So it can be represented as C1
times V1 plus C2 times V2 all the way to plus Ck
times with Vk. Let me use a different letter
just to make sure that you understand that this is a
different equation that I'm writing than I wrote earlier
in the video. So I can write this, and then
I can have some other constants that say plus e1 times
our V perp basis vector 1 plus e2 times this guy plus
all the way to en minus k times the n minus k basis
vector for V perp. I can represent any vector
in Rn this way. Or another way to say it. What is this? This is some vector that is a
member of our subspace V. And then this is some vector
over here that is a member of the orthogonal complement
of V. This is just a linear
combination of V perp's basis vectors. This is just a linear
combination of V's basis vectors. So given that all of these
characters are a basis for Rn tells us that any member of Rn
can be represented as a linear combination of them. But that means that any member
of Rn can be represented as a sum of a member of our subspace
V plus a member of your subspace V perp. This is a member of V, and this
is a member of V perp. And that's a really, really
interesting idea. You give me a subspace and then
we can figure out its orthogonal complement. Any other vector in Rn can be
represented as a combination, or sum, of some vector in our
subspace and some vector in its orthogonal complement. Now the next question you
might be asking, is this representation unique? So is this unique? Well let's test it out by
assuming it's not unique. So that means that I have for
some vector a that is a member of Rn, I can represent
it two ways. I can represent it as equalling
some member of my subspace V, plus some member
of the orthogonal complement of V. I can represent it that way. Or I could represent it as
some other member of my subspace V plus some other
member of my orthogonal complement. So x1, x2 are members
of V perp. And then v1 and v2
are members of V. If we assume it's not unique,
there's two ways that I could do this. And I'm representing it
as these two vectors. Now clearly this side of this
equation is equal to that. These are both representations
of a. So we can rearrange
this a little bit. We could say that v1 minus v2--
if I subtract v2 from both sides, I get v1 minus
v2 is equal to-- that's subtracting v2 from both sides,
and if I subtract x1 from both sides-- x2 minus x1. These are both members
of the subspace V. And any subspace is closed
under addition and subtraction, which is really
a special case of addition. The vector v1, let me write it
this way, let me call my sum vector z, being equal to both of
these guys which are equal to each other. z is the vector V1 minus V2. Any subspace is closed under
addition, if you take two vectors and find their
difference in a subspace, then that resulting difference
is also going to be in a subspace. z is going to be a member
of our subspace V. This vector right here-- which
is also the same thing we just said that to be equal our vector
z-- is going to be a member of our V perp. Why? Because both x1 and x2 are
members of the subspace V's orthogonal complement. And that is a subspace
as well. So it is closed under addition
and subtraction. So this is also going to be
a member of your subspace. So we could also say that z is
a member of V perp or the orthogonal complement of V. Well, we've done this
multiple times. This was the first thing
we showed in the video. The only vector that's a member
of a subspace and its orthogonal complement
is the 0 vector. So z has to be equal
to the 0 vector. So this is equal to
the 0 vector. Well, if both of these are equal
to the 0 vector, we know that v1 minus v2 is equal to
the 0 vector, which implies that v1 must be equal to v2. And we also know that x2 minus
x1 is equal to the 0 vector. Or x2 is equal to x1. So we try to say that, hey,
there's two ways to construct some arbitrary vector
a that's in Rn. And we wrote that down. But then we found out that no,
v1 must be equal to v2 and x1 must be equal x2. So there's only a unique way to
write any member of Rn as a sum of a vector that's in our
subspace V and a vector that is in the orthogonal
complement of V.