It is my experience that proofs involving matrices can be shortened by 50% if one throws matrices out.
-- Emil Artin Hey everyone! Where we last left off, I showed what linear
transformations look like and how to represent them using matrices. This is worth a quick recap, because it's
just really important. But of course, if this feels like more than
just a recap, go back and watch the full video. Technically speaking, linear transformations
are functions, with vectors as inputs and vectors as outputs. But I showed last time how we can think about
them visually as smooshing around space in such a way the gridlines
stay parallel and evenly spaced, and so that the origin remains fixed. The key take-away was that a linear transformation is completely determined,
by where it takes the basis vectors of the space which, for two dimensions, means i-hat and
j-hat. This is because any other vector can be described
as a linear combination of those basis vectors. A vector with coordinates (x, y) is x times i-hat + y times j-hat. After going through the transformation this property, the grid lines remain parallel
and evenly spaced, has a wonderful consequence. The place where your vector lands will be
x times the transformed version of i-hat + y times the transformed version of j-hat. This means if you keep a record of the coordinates
where i-hat lands and the coordinates where j-hat lands you can compute that a vector which starts
at (x, y), must land on x times the new coordinates of
i-hat + y times the new coordinates of j-hat. The convention is to record the coordinates
of where i-hat and j-hat land as the columns of a matrix and to define this sum of the scaled versions
of those columns by x and y to be matrix-vector multiplication. In this way, a matrix represents a specific linear transformation and multiplying a matrix by a vector is, what
it means computationally, to apply that transformation to that vector. Alright, recap over. Onto the new stuff. Often-times you find yourself wanting to describe
the effect of applying one transformation and then another. For example, maybe you want to describe what happens when
you first rotate the plane 90° counterclockwise then apply a shear. The overall effect here, from start to finish, is another linear transformation, distinct from the rotation and the shear. This new linear transformation is commonly called the “composition” of the two separate transformations we applied. And like any linear transformation it can be described with a matrix all of its
own, by following i-hat and j-hat. In this example, the ultimate landing spot
for i-hat after both transformations is (1, 1). So let's make that the first column of the
matrix. Likewise, j-hat ultimately ends up at the
location (-1, 0), so we make that the second column of the matrix. This new matrix captures the overall effect
of applying a rotation then a sheer but as one single action, rather than two
successive ones. Here's one way to think about that new matrix: if you were to take some vector and pump it
through the rotation then the sheer the long way to compute where it ends up is to, first, multiply it on the left by the
rotation matrix; then, take whatever you get and multiply that
on the left by the sheer matrix. This is, numerically speaking, what it means to apply a rotation then a sheer
to a given vector. But, whatever you get should be the same as
just applying this new composition matrix that we just found, by
that same vector, no matter what vector you chose, since this new matrix is supposed to capture
the same overall effect as the rotation-then-sheer action. Based on how things are written down here I think it's reasonable to call this new matrix,
the “product” of the original two matrices. Don't you? We can think about how to compute that product
more generally in just a moment, but it's way too easy to get lost in the forest
of numbers. Always remember, the multiplying two matrices
like this has the geometric meaning of applying one
transformation then another. One thing that's kinda weird here, is that
this has reading from right to left; you first apply the transformation represented
by the matrix on the right. Then you apply the transformation represented
by the matrix on the left. This stems from function notation, since we write functions on the left of variables, so every time you compose two functions, you
always have to read it right to left. Good news for the Hebrew readers, bad news
for the rest of us. Let's look at another example. Take the matrix with columns (1, 1) and (-2, 0) whose transformation looks like this, and let's call it M1. Next, take the matrix with columns (0, 1)
and (2, 0) whose transformation looks like this, and let's call that guy M2. The total effect of applying M1 then M2 gives us a new transformation. So let's find its matrix. But this time, let's see if we can do it without
watching the animations and instead, just using the numerical entries
in each matrix. First, we need to figure out where i-hat goes after applying M1 the new coordinates of i-hat, by definition, are given by that first column
of M1, namely, (1, 1) to see what happens after applying M2 multiply the matrix for M2 by that vector
(1,1). Working it out, the way that I described last
video you'll get the vector (2, 1). This will be the first column of the composition
matrix. Likewise, to follow j-hat the second column of M1 tells us the first
lands on (-2, 0) then, when we apply M2 to that vector you can work out the matrix-vector product
to get (0, -2) which becomes the second column of our composition
matrix. Let me talk to that same process again, but
this time, I'll show variable entries in each matrix, just to show that the same line of reasoning
works for any matrices. This is more symbol heavy and will require
some more room, but it should be pretty satisfying for anyone
who has previously been taught matrix multiplication the more rote way. To follow where i-hat goes start by looking at the first column of the
matrix on the right, since this is where i-hat initially lands. Multiplying that column by the matrix on the
left, is how you can tell where the intermediate
version of i-hat ends up after applying the second transformation. So, the first column of the composition matrix will always equal the left matrix times the
first column of the right matrix. Likewise, j-hat will always initially land
on the second column of the right matrix. So multiplying the left matrix by this second
column will give its final location and hence, that's the second column of the
composition matrix. Notice, there's a lot of symbols here and it's common to be taught this formula
as something to memorize along with a certain algorithmic process to
kind of help remember it. But I really do think that before memorizing
that process you should get in the habit of thinking about
what matrix multiplication really represents: applying one transformation after another. Trust me, this will give you a much better
conceptual framework that makes the properties of matrix multiplication
much easier to understand. For example, here's a question: Does it matter what order we put the two matrices
in when we multiply them? Well, let's think through a simple example like the one from earlier: Take a shear which fixes i-hat and smooshes
j-hat over to the right and a 90° rotation. If you first do the shear then rotate, we can see that i-hat ends up at (0, 1) and j-hat ends up at (-1, 1) both are generally pointing close together. If you first rotate then do the shear i-hat ends up over at (1, 1) and j-hat is off on a different direction
at (-1, 0) and they're pointing, you know, farther apart. The overall effect here is clearly different so, evidently, order totally does matter. Notice, by thinking in terms of transformations that's the kind of thing that you can do in
your head, by visualizing. No matrix multiplication necessary. I remember when I first took linear algebra there's this one homework problem that asked
us to prove that matrix multiplication is associative. This means that if you have three matrices
A, B and C, and you multiply them all together, it shouldn't matter if you first compute A
times B then multiply the result by C, or if you first multiply B times C then multiply
that result by A on the left. In other words, it doesn't matter where you
put the parentheses. Now if you try to work through this numerically like I did back then, it's horrible, just horrible, and unenlightening
for that matter. But when you think about matrix multiplication
as applying one transformation after another, this property is just trivial. Can you see why? What it's saying is that if you first apply
C then B, then A, it's the same as applying C, then B then A. I mean, there's nothing to prove, you're just applying the same three things
one after the other all in the same order. This might feel like cheating. But it's not! This is an honest-to-goodness proof that matrix
multiplication is associative, and even better than that, it's a good explanation
for why that property should be true. I really do encourage you to play around more
with this idea imagining two different transformations thinking about what happens when you apply
one after the other and then working out the matrix product numerically. Trust me, this is the kind of play time that
really makes the idea sink in. In the next video I'll start talking about
extending these ideas beyond just two dimensions. See you then!
