If we think about a matrix as a transformation of space it can lead to a deeper understanding of matrix operations. This viewpoint helps motivate how we define matrix operations like multiplication, and, it gives us a nice excuse to
draw pretty pictures. This material touches on linear algebra (usually a college topic).
Multiplication as a transformation
The idea of a "transformation" can seem more complicated than it really is at first, so before diving into how 2×22, times, 2 matrices transform 22-dimensional space, or how 3×33, times, 3 matrices transform 33-dimensional space, let's go over how plain old numbers (a.k.a. 1×11, times, 1 matrices) can be considered transformations of 11-dimensional space.
"11-dimensional space" is simply the number line.
What happens when you multiply every number on the line by a particular value, like 22? One way to visualize this is as follows:
We keep a copy of the original line for reference, then slide each number on the line to 22 times that number.
Similarly, multiplication by 21start fraction, 1, divided by, 2, end fraction could be visualized like this:
And so that negative numbers don't feel neglected, here is multiplication by −3minus, 3:
For those of you fond of fancy terminology, these animated actions could be described as "Linear transformations of 11-dimensional space". The word “transformation” means the same thing as “function”: something which takes in a number and outputs a number, like f(x)=2xf, left parenthesis, x, right parenthesis, equals, 2, x. However, while we typically visualize functions with their graphs, people tend to use the word “transformation” to indicate that you should instead visualize some object moving, stretching, squishing, etc. So the function f(x)=2xf, left parenthesis, x, right parenthesis, equals, 2, x visualized as a transformation gives us the "Multiplication by 22" video above. It moves the point 11 on the number line to where 22 starts off, moves 22 to where 44 starts off, etc.
The technical definition of a “linear” transform is a function f(x)f, left parenthesis, x, right parenthesis that satisfies the two properties
f(x+y)=f(x)+f(y)f, left parenthesis, x, plus, y, right parenthesis, equals, f, left parenthesis, x, right parenthesis, plus, f, left parenthesis, y, right parenthesis
f(cx)=cf(x)f, left parenthesis, c, x, right parenthesis, equals, c, f, left parenthesis, x, right parenthesis
Here, in 1 dimension, xx and yy are numbers, as opposed to, say, vectors. In this special case of 1 dimension, the only functions satisfying these properties look like multiplication by a constant, i.e. f(x)=kxf, left parenthesis, x, right parenthesis, equals, k, x for some constant kk.
Why? Plug in x=1x, equals, 1 to the second property, and we see f(c)=cf(1)f, left parenthesis, c, right parenthesis, equals, c, f, left parenthesis, 1, right parenthesis. If we interpret cc as a variable and f(1)f, left parenthesis, 1, right parenthesis as some constant, the entire function is just multiplication by f(1)f, left parenthesis, 1, right parenthesis. For instance, if f(1)=8f, left parenthesis, 1, right parenthesis, equals, 8, then f(x)=8xf, left parenthesis, x, right parenthesis, equals, 8, x. In fact, we don't even need the first property, since once we know ff looks like f(x)=kxf, left parenthesis, x, right parenthesis, equals, k, x, the fact that f(x+y)=f(x)+f(y)f, left parenthesis, x, plus, y, right parenthesis, equals, f, left parenthesis, x, right parenthesis, plus, f, left parenthesis, y, right parenthesis follows from the distributive law: k(x+y)=kx+kyk, left parenthesis, x, plus, y, right parenthesis, equals, k, x, plus, k, y.
This might seem like an awfully complicated way to describe multiplication, especially given that the first property above is pointless. However, the importance of linearity comes when ff is a function of vectors. The fact that in 11-dimension ff is completely determined by where it takes the number 11 has a much more interesting analog in higher dimensions.
Before we move on to 22-dimensional space, there's one simple but important fact we should keep in the back of our minds. Suppose you watch one of these transformations, knowing that it's multiplication by some number, but without knowing what that number is, like this one:
You can easily figure out which number is being multiplied into the line by following1start color goldE, f, o, l, l, o, w, i, n, g, space, 1, end color goldE. In this case, 11 lands where −3minus, 3 started off, so you can tell that the animation represents multiplication by −3minus, 3.
Thinking of numbers as transformations gives an alternate interpretation of multiplication.
If we apply two transformation in a row, for instance multiplication by 22, then by 33,
the total action is the same as some other single transformation, in this case multiplication by 66:
This gives a convoluted understanding of multiplication, which will be surprisingly useful as an analogy for matrix multiplication.
Person A: What is 4×54, times, 5?
Person B: Well, consider the unique linear transform which takes 11 to 44, as well as the unique linear transform which takes 11 to 55, then apply each of these transformations, one after the other, and 4×54, times, 5 will be whatever number 11 lands on.
Person A: ...that's the stupidest thing I've ever heard.
What do linear transformations in 22 dimensions look like?
A 22-dimensional linear transformation is a special kind of function which takes in a 22-dimensional vector [xy] and outputs another 22-dimensional vector. As before, our use of the word “transformation” indicates we should think about smooshing something around, which in this case is 22-dimensional space. Here are some examples:
For our purposes, what makes a transformation linear is the following geometric rule: The origin must remain fixed, and all lines must remain lines. So all the transforms in the above animation are examples, but the following are not:
As in 1 dimension, what makes a transformation “linear” is that it satisfies the two properties
where and are now vectors instead of numbers. While in 1-dimension the first property was useless, it now plays a more important role, because in some sense it determines how the two different dimensions play together during a transformation.
Following specific vectors during a transformation
Imagine you are watching one particular transformation, like this one
How could you describe this to a friend who is not watching the same animation? You can no longer describe it using a single number, the way we could just follow the number 11 in the one dimensional case. To help keep track of everything, let's put a green arrow over the vector
put a red arrow over the vector
and fix a copy of the grid in the background.
Now it's a lot easier to see where things land. For example, watch the animation again, and focus on the vector , we can more easily follow it to see that it lands on the vector [4−2].
We can represent this fact with the following notation:
Practice Problem: Where does the point at [−10] end up after the plane has undergone the transformation in the above video?
Practice Problem Even though it has gone off screen, can you predict where the point  has landed?
Notice, a vector like , which starts off as 22 times the green arrow, continues to be 22 times the green arrow after the transformation. Since the green arrow lands on [1−2], we can deduce that
And in general
Similarly, the destination of the entire yy-axis is determined by where the red arrow
lands, which for this transformation is .
Practice Problem: After the plane has undergone the transformation illustrated above, where does the general point [0y] on the yy-axis land?
In fact, once we know where
land, we can deduce where every point on the plane must go. For example, let's follow the point
in our animation:
It starts at −1minus, 1 times the green arrow plus 22 times the red arrow, but it also ends at −1minus, 1 times the green arrow plus 22 times the red arrow, which after the transformation means
This ability to break up a vector in terms of its components both before and after the transformation is what's so special about linear transformations.
Practice Problem: Use this same tactic to compute where the vector [1−1] lands.
Representing two dimensional linear transforms with matrices
In general, since each vector
can be broken down as
If the green arrow
lands on some vector
and the red arrow
lands on some vector
then the vector
must land on
A really nice way to describe all this is to represent a given linear transform with the matrix
where the first column tells us where
lands and the second column tells us where
lands. Now we can describe where any vector
lands very compactly as the matrix-vector product
In fact, this is where the definition of a matrix-vector product comes from.
So in the same way that 11-dimensional linear transforms could be described as multiplication by some number, namely whichever number 11 lands on top of, 22-dimensional linear transforms can always be described by a 2×22, times, 2matrix, namely the one whose first column indicates where  lands, and whose second column indicates where  lands.