Let's say I have three position vectors here in R2.
Let me scroll this over a little bit.
Let's say my first position vector is x0 and it is equal
to minus 2, minus 2.
So if I were to graph x0 I would go minus 2, minus 2.
x0 looks like that.
My next position vector I have is x1 and I'll say that's
equal to minus 2, 2.
If I were to graph here, minus 2, 2.
That's my next position vector right there.
And this is x1.
And when I say it's a position vector, they specify a
specific coordinate in R2.
Let me draw a third one just for fun, x2.
Let's say that that is equal to 2, minus 2.
So if I were to draw this, 2 minus 2, it goes right here.
So that vector right there is x2.
Now, what I'm curious about or, I
guess, not curious about.
What I want to do here is define the line segments that
connect these points.
So let's say I have my first line segment.
Let me call it L1, or let me call it L0.
And I want it to be the line segment that
connects x0 to x1.
How can I construct that?
So I want to construct this line segment right here, this
little-- let me do it in a different color actually.
Let me do it in orange, L0.
So what I want to do is I want to find the set of all of
these values right here, all of the position vectors that
define points on this line right there.
Well we could define it as, we could start off at x0.
We could say that orange line is x0 plus scaled versions of
the difference of x1 and x0.
If you take x1 minus x0, you get this vector right there.
That's x1 minus x0.
That orange vector.
I know I wrote it right there so it's hard to read.
But if you just take x1 minus x0 you get that and that make
sense. x0 plus this orange vector is equal
to that blue factor.
So if you just take different scaled up versions of this
guy, you're going to end up at different
points in this direction.
We're starting at x0, maybe I should do that in green.
We're starting at x0 and then we're going to add up scaled
up versions of this orange vector, which is just the
difference of x1 and x0.
Let me write that.
So scaled up versions of x1 minus x0.
Now, we have to constrict.
If we just want to be in this line segment, we have to
constrict Rt.
If we said t was a member of real numbers, if it was any
real number, then we would essentially be defining the
set of this entire vertical line going up and down to
infinity in the upwards direction and
the downwards direction.
But we just want to restrict it to start here
and then go up here.
And it doesn't necessarily have to have any direction.
We could say this is true, our little line segment here is
true, for t-- let me write it this way.
t is greater than or equal to 0.
So when t equals 0, this term cancels out and we just have
this point or this position right here.
Let me draw it in green.
We just have that position there.
And then t is going to be less than or equal to 1.
What happens when t is equal to 1?
When t is equal to 1, this becomes x1 minus x0.
You have an x0 there.
This x0 and that x0 cancel out and you're just at this point
right there.
When t is equal to 1/2, just to make sure this all make
sense to you, what happens?
You have x1 minus x0, which is this orange vector right here.
When t is equal to 1/2 you're essentially scaling that
orange vector by half and you end up right at that point,
which is exactly where you want to be.
You want halfway along that line segment.
At t is equal to 0.25, you're going to be here.
t is equal to 0.75, you're going to be there.
So at any value for t being any real number between 0 and
1, you're going to end up at all of the points along that
line segment.
So that's our L0.
It's just a set of vectors.
Now we can do the same exercise if we wanted to find
out the line, the equation of the line, that goes
between x1 and x2.
If we wanted to find the equation of that line, we
could call this L1.
And L1 would be equal to x1 plus t times x2 minus x1 for 0
is less than or equal to t is less than or equal to 1.
That's L1.
And then finally, if we want to make a triangle out of
this, let's define this line right here.
Let's define that as L2.
L2 would be equal to the set of all of vectors where you
start off at x2.
Set of all of vectors that are x2 plus some scaled up sum of
x0 minus x2.
x0 minus x2 is this vector right here.
So x0 minus x2 such that 0 is less than or equal to t is
less than or equal to 1.
And so if you take the combination, if you were to
define kind of a super set-- I could have defined my shape
as-- let's say it's the union of all of those guys.
Well, let me just write it.
L0, L1, and L2.
Then you'd have a nice triangle here.
If you take the union of all of these three sets, you get
that nice triangle there.
Now, what I want to do in this video, I think this is all a
bit of review for you.
But it's maybe a different way of looking at things than
we've done in the past. Is I want to understand what
happens to this set right here when I take a transformation,
a linear transformation, of it?
So let me define a transformation.
I'll make it a fairly straightforward
transformation.
Let me define my transformation of x, of any x,
to be equal to the matrix 1 minus 1, 2, 0 times
whatever vector x.
So times x1, x2.
And we know that any linear transformation can actually be
written as a matrix and vice versa.
So you might have said, hey, you know, you're giving an
example with the matrix, what about all those other ways to
write in your transformation?
You can write all of those as a matrix.
So let's translate-- let's try to figure out what this is
going to look like.
What our triangle is going to look like when we transform
every point in it.
Let me take the transformation first. The transformation of
L0 is equal to the transformation of this thing.
This is just one of the particular members.
For a particular t, this is one of the
particular members of L0.
So it's going to be equal to the transformation of x0 minus
the transformation of x1 minus x0 such that-- sorry.
Minus t times x1 minus x0.
That's a lowercase t, not the transformation.
Such that 0 is less than or equal to t is less than or
equal to 1.
Let me switch colors.
This, just by the properties of linear transformations,
this is equal to the transformation-- let me put
the brackets out-- of x0 minus the transformation of our
scalar t times x1 minus x0 for all t's between 0 and 1.
That part is getting a little redundant to keep saying it.
And then, what does this equal to?
If you I take the transformation of a scaled up
vector, that's just the scaled up
transformation of that vector.
So this is going to be equal to this part, the
transformation of x0 minus t, our scalar multiple t, times
the transformation of the vector x1 minus x0.
And then let me make sure I get my parentheses right.
Such that 0 is less than or equal to t is less than or
equal to 1.
And then the transformation of the sum of two vectors is
equal to the sum of their transformations.
We've all seen this before.
So our transformation of our first line-- this one right
here-- L0, is equal to the set where it's the transformation
of x0 minus t times the transformation of x1 minus the
transformation of x0.
And we've just done our first line so far.
I have a parentheses there.
For 0 is less than t is less than or equal to 1.
Now this is a pretty neat result and it's going to
simplify our lives a lot.
The transformation of the line segment that goes from x0 to
x1 ends up just being the line segment that goes from the
transformation of x0 to the transformation of x1.
Let me make this clear.
What is the transformation of x0?
Let's calculate these things.
So x0 was minus 2, minus 2.
Let me write out the transformation of x0.
So the transformation of x0 is equal to-- let me write it
out, so I don't make any careless mistakes.
1, 2, minus 1, 0 times minus 2 minus, minus 2.
And so what would this be equal to?
1 minus 2 minus-- so it's 1 minus 2 plus minus
1 times minus 2.
That's going to be plus 2.
So it's minus 2 plus a plus 2, so it's equal to 0.
And then we have 2 times minus 2, which is minus 4.
2 times minus 4.
And then plus 0 times, so it's minus 4.
So that's the transformation of x0.
Let me graph it.
So it's 0 minus 4.
So our x0 vector.
So this is the transformation of x0.
So the transformation associated this vector with
this vector down here, the one that goes straight down.
Now let me take the transformation
of the other guys.
The transformation of x1.
I'll just do it right here.
I'm running out of space.
The transformation of x1 is equal to 1, 2, minus 1, 0
times minus 2, 2.
So what is this equal to?
This is equal to 1 times minus 2, plus minus 1 times 2.
So that's minus 4.
And then 2 times minus 2 is minus 4.
Plus 0.
So minus 4, minus 4.
So x1 is minus 4, minus 4.
So our x1 looks like this.
Our transformation of x1 is this vector right here in R2.
Our transformation is going from R2 to R2.
So that's why I'm able to draw them both on this nice
Cartesian coordinate plane.
And we have one left.
Let's take our transformation of x2.
So the transformation of x2 is equal to our transformation
matrix 1, 2, minus 1, 0 times 2, minus 2.
And so this will be equal to 1 times 2 is 2.
Plus minus 1 times minus 2.
So it's 2 plus 2 is 4.
And then have 2 times 2 is 4.
Plus 0 times minus 2.
So it's 4, 4.
So x2 is 4, 4.
So it's this point right here.
4, 4 right there.
So the transformation of x2 is that vector right there.
And so, we are able to take the transformation of each of
these points of this triangle.
But who knows what the transformation does to
everything in between, to all of these other-- the actual
sides of the triangle.
We're able to do a little math and we just
did the first side.
We just did L0 right there we found, just using the
properties of a linear transformation, the definition
of a linear transformation actually, we were able to find
that the transformation of L0 of this vertical line here, it
just ends up becoming the line where we can start off at the
transformation of x0.
The point specified by this vector right here.
And to that I add up scaled multiples of the
transformation of x1 minus the transformation of x0.
What is this, the transformation of x1 minus the
transformation of x0?
The transformation of x1 is just this vector right here.
The transformation of x0 is just that vector.
So this whole term right here is just this vector minus that
vectors or it's this vector right there.
It's just that vector right there.
And so, what we essentially have, we've defined the same
way that we did in the first part of this video.
This is just the same thing as the line segment that connects
the point defined here and the point defined there.
We took the difference of the two and we have scaled up
versions of that between t is equal to 0 and 1.
So the transformation of L0 really just became the
transformation-- is just the line between the
transformations of both of the endpoints, which is a pretty
neat result.
It makes our lives simple.
We can do the exact same logic to say, you know what?
What's going to be the transformation of L1?
Well L1 was between the points x1 and x2.
It was between that point and that point.
That was L1.
So using the same logic, we can do the
math all over again.
But it applies to any line.
I did it all abstractly here.
The transformation of L1 is going to be the line that
connects the transformation of the two endpoints.
So it's going to be the line that connects the
transformation of x1 and the transformation of x2.
Let me make this right here, is the transformation of L1.
This right here is the transformation of L0.
And then finally, what's the transformation of L2?
L2 connects the points x2 and x0.
So that is L2 right there.
So the transformation of it using the same math that we've
done before is really just the line connecting the
transformations of those two points.
So the transformation of L2 is going to be equal to the line
that connects the transformation of x2 to the
transformation of x0.
So it's going to be that line right there.
So this is the transformation of L2, or if we defined our
whole shape or our whole triangle as the set of all of
these, the transformation of that.
So the transformation of our whole shape is
now this skewed triangle.
I think you're now getting a sense of why this might be
useful in things like computer graphics or game development.
Because when you look at things from different angles,
you start to skew them and whatever else.
But taking this transformation, we were able
to turn this set of vectors or the positions-- or I guess
this shape, which is specified by this set of
vectors right here.
We were able to change it into this shape in R2 specified by
a different set of vectors.
But the whole takeaway from this video is, you don't have
to individually figure out, gee, what does this point
right here translate into over here?
All you have to say is what were my endpoints?
Figure out their transformations and then
connect the dots in the same order.
That's what is essentially the big takeaway from here.
And this idea of when you transform one set into another
set, they have some terminology
associated with it.
So let's say the transformation of L0.
L0 was the set of vectors that specified
this line right here.
The transformation of L0, which is this set, is a set of
vectors-- sorry.
L0 was this set.
It was the set of vectors in our co-domain that specified
these points.
This is called the image of L0 under T.
And it kind of makes sense.
Why do they call it the image?
Because t is taking this thing right here, this L0, and kind
of distorting it or creating a new image
of it in the co-domain.
It's taking a set in the domain and creating a new
image of it in the co-domain right there.
We could say that t of the transformation of our entire
shape, I defined our entire shape up here as this whole
triangle right there.
That's the image and that turned into this purple
triangle here.
That's the image of s under T.
So hopefully you found that pretty interesting.
This is actually a super useful takeaway if you ever
want to become a 3D programmer of some type.
In then next video, we'll explore what happens when s is
no longer just a subset of our domain.
Everything we've been dealing so far-- L0, L1, and L2, or
our entire triangle, these were all subsets or Rn.
In the next video we'll talk about what happens when you
take the transformation of all of Rn.