Video Transcript
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
full 3D grid representing all points can get kinda 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 three-by-three matrix. This gives a matrix that completely
describes the transformation using only nine numbers. As a simple example, consider the
transformation that rotates space 90 degrees around the 𝑦-axis. So that would mean that it takes
𝑖-hat to the coordinates zero, zero, negative one on the 𝑧-axis. It doesn’t move 𝑗-hat so it stays
at the coordinates zero, one, zero. And then 𝑘-hat moves over to the
𝑥-axis at one, zero, zero. 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 2D 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 three-by-three
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 3D 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 transformations in space.
In the next video, I’ll start
getting into the determinant.
