[classical music]
"Lisa: Well, where's my dad?
Frink: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology that Homer Simpson has stumbled into
... (dramatic pause) ...
the third dimension."
Hey folks I've got a relatively quick video for you today,
just sort of a footnote between chapters.
In the last two videos I talked about
linear transformations and matrices, but, I only showed the specific case of
transformations that take two-dimensional vectors to other
two-dimensional vectors.
In general throughout the series we'll work mainly
in two dimensions.
Mostly because it's easier to actually see on the screen and wrap your mind around,
but, more importantly than that
once you get all the core ideas in two dimensions they carry over pretty
seamlessly to higher dimensions.
Nevertheless it's good to peak our heads outside of flatland now and then to...
you know see what it means to apply these ideas in more than just those two dimensions.
For example, consider a linear transformation with three-dimensional vectors as inputs
and three-dimensional vectors as outputs.
We can visualize this by smooshing around all the points in three-dimensional space,
as represented by a grid, in such a way that keeps the grid lines parallel
and evenly spaced and which fixes the origin in place.
And just as with two dimensions, every point of space that we see moving around
is really just a proxy for a vector who has its tip at that point,
and what we're really doing is thinking about input vectors
*moving over* to their corresponding outputs,
and just as with two dimensions,
one of these transformations is completely described by where the basis vectors go.
But now, there are three standard basis vectors that we typically use:
the unit vector in the x-direction, i-hat;
the unit vector in the y-direction, j-hat;
and a new guy—the unit vector in the z-direction called k-hat.
In fact, I think it's easier to think about these transformations
by only following those basis vectors
since, the for 3-D grid representing all points can get kind of messy
By leaving a copy of the original axes in the background,
we can think about the coordinates of where each of these three basis vectors lands.
Record the coordinates of these three vectors as the columns of a 3×3 matrix.
This gives a matrix that completely describes the transformation using only nine numbers.
As a simple example, consider, the transformation that rotate space
90 degrees around the y-axis.
So that would mean that it takes i-hat
to the coordinates [0,0,-1] on the z-axis,
it doesn't move j-hat so it stays at the coordinates [0,1,0]
and then k-hat moves over to the x-axis at [1,0,0].
Those three sets of coordinates become the columns of a matrix
that describes that rotation transformation.
To see where vector with coordinates XYZ lands the reasoning is almost identical
to what it was for two dimensions—each of those coordinates can be thought of
as instructions for how to scale
each basis vector so that they add together to get your vector.
And the important part just like the 2-D case is that this scaling and adding process
works both before and after the transformation.
So, to see where your vector lands you multiply those coordinates
by the corresponding columns of the matrix and
then you add together the three results.
Multiplying two matrices is also similar
whenever you see two 3×3 matrices getting multiplied together
you should imagine first applying the transformation encoded by the right one
then applying the transformation encoded by the left one.
It turns out that 3-D matrix multiplication is actually pretty
important for fields like computer graphics and robotics—since things like
rotations in three dimensions can be pretty hard to describe, but,
they're easier to wrap your mind around if you can break them down as the composition
of separate easier to think about rotations
Performing this matrix multiplication numerically, is, once again pretty similar
to the two-dimensional case.
In fact a good way to test your understanding of
the last video would be to try to reason through what specifically this matrix
multiplication should look like thinking closely about how it relates to the idea
of applying two successive of transformations in space.
In the next video I'll start getting into the determinant.