[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.