Linear Alg: Visualizing a projection onto a plane

I'm going to do one more video
where we compare old and new

definitions of a projection.

Our old definition of a
projection onto some line, l,

of the vector, x, is the vector
in l, or that's a

member of l, such that x minus
that vector, minus the

projection onto l of x,
is orthogonal to l.

So the visualization is, if
you have your line l like

this, that is your line
l right there.

And then you have some other
vector x that we're take the

projection of it on to l.

So that's x.

The projection of x onto l,
this thing right here, is

going to be some vector in l.

Such that when I take the
difference between x and that

vector, it's going to
be orthogonal to l.

So it's going to be
some vector in l.

This was our old definition when
we took the projection

onto a line.

Some vector in l.

Maybe it's there.

And if I take the difference
between that and that, this

difference vector's going
to be orthogonal to

everything in l.

Just like that.

So, this right here would be
it's difference vector.

That would be x minus the
projection of x onto l.

And then, of course, this
vector right here.

This is the one we
were defining it.

That was the projection
onto l of x.

Now, what's a different
way that we could

have written this?

We could have written this
exact same definition.

We could have said it is the
vector in l such that-- so we

could say, let me write
it here in purple.

Is the vector v in l such that
v-- let me write it this way--

such that x minus v, right?

x minus the projection of l is
orthogonal is equal to w,

which is orthogonal to
everything in l.

Being orthogonal to l literally
means being

orthogonal to every
vector in l.

So I just rewrote it a little
bit different, instead of just

leaving it as a projection
of x onto l.

I said hey, that's some vector,
v, in l such that x

minus v is equal to some other
vector, w, which is orthogonal

to everything in l.

Or we can rewrite that statement
right there as x is

equal to v plus w.

So we can just say that the
projection of x onto l is the

unique vector v in l, such that
x is equal to v plus w,

where w is a unique vector-- I
mean it is going to be unique

vector-- in the orthogonal
complement of l.

Right?

This is got to be orthogonal
to everything in l.

So that's going to be a member
of the orthogonal

complement of l.

So this definition is actually
completely consistent with our

new subspace definition.

And we could just extend
it to arbitrary

subspaces, not just lines.

Let me help you visualize
that.

So let's say we're dealing
with R3 right here.

And I've got some
subspace in R3.

And let's say that subspace
happens to be a plane.

I'm going to make it a plane
just so that it becomes clear

that we don't have to take
projections just onto lines.

So this is my subspace
v right there.

Let me draw its orthogonal
compliment.

Let's say its orthogonal
complement looks

something like that.

Let's say it's a line.

And then it goes-- it intersects
right there.

Then it goes back.

And, of course, it would have to
intersect at the 0 vector.

That's the only place where a
subspace and its orthogonal

complement overlap.

And then it goes behind
and you see it again.

Obviously you wouldn't be able
to you again because this

plane would extend in
every direction.

But you get the idea.

So this right here
is the orthogonal

complement of v, that line.

Now, let's have some other
arbitrary vector in R3 here.

So let's say I have some vector
that looks like that.

Let's say that that is x.

Now our new definition for the
projection of x onto v is

equal to the unique vector v.

This is a vector v.

That's a subspace v.

The unique vector v, that is a
member of v, such that x is

equal to v plus w, where w
is a unique member of the

orthogonal complement of v.

This is our new definition.

So, if we say x is equal to
some member of v and some

member of its orthogonal
complement-- we can visually

understand that here.

We could say, OK it's going to
be equal to, on v, it'll be

equal to that vector
to right there.

And then on v's orthogonal
complement,

you add that to it.

So, if you were to shift it
over, you would get that

vector, just like that.

This right here is v.

That right there is v.

And then this is vector that
goes up like this, out of the

plane, orthogonal to
the plane, is w.

You could see if you take v plus
w, you're going to get x.

And you could see that v is
the projection onto the

subspace capital v-- so this
is a vector, v-- is the

projection onto the subspace
capital V of the vector x.

So the analogy to a shadow
still holds.

If you imagine kind of a light
source coming straight down

onto our subspace, kind of
orthogonal to our subspace,

the projection onto our subspace
is kind of the shadow

of our vector x.

Hopefully that help you
visualize it a little better.

But what we're doing here is
we're going to generalize it.

Earlier in this video
I showed you a line.

This is a plane.

But we can generalize
it to any subspace.

This is in R3.

We can generalize it
to Rn, to R100.

And that's really the power
of what we're doing here.

It's easy to visualize it here,
but it's not so easy to

visualize it once you get
to higher dimensions.

And actually, one other thing.

Let me show that this new
definition is pretty much

almost identical to exactly
what we did with lines.

This is identical to saying that
the projection onto the

subspace x is equal to some
unique vector in V such that x

minus the projection onto v of
x is orthogonal to every

member of V.

Because this statement, right
here, is saying any vector

that's orthogonal to any member
of v says that it's a

member of the orthogonal
complement of v.

So that statement could be
written as x minus the

projection onto v of x is a
member of v's orthogonal

complement.

Or we could call some w.

So if you call this your v,
and if you call this whole

thing your w, you get this exact
definition right there.

You would have w is equal
to x minus v.

And then if you add v to both
sides, you get w plus v is

equal to x.

We defined v to be, the
orthogonal-- the

projection of x onto v.

w is a member of our orthogonal
complement.

And I don't want you
to get confused.

The vector v is the orthogonal
projection of our vector x

onto the subspace capital V.

I probably should use different
letters instead of

using a lowercase and
a uppercase v.

It makes the language
a little difficult.

But I just wanted to give you
another video to give you a

visualization of projections
onto

subspaces other than lines.

And to show you that our old
definition, with just a

projection onto a line which was
a linear transformation,

is essentially equivalent
to this new definition.

On the next video, I'll show
you that this, for any

subspace is, indeed, a linear
transformation.