This video is sponsored by Brilliant.org and it’s the second video in a mini series about

Linear Algebra. Click here and watch the first one on vectors and bases for vector spaces.

This video is about Matrices. When I first met a matrix it was in a textbook question.

It asked me to put a bunch of numbers to do with a bakery’s sales in an array. I was

very confused. It took me a long long time to really understand that matrices have really

nothing to do with bakeries and that just the same way that writing a vector as a column

of numbers is just a shorthand, writing a matrix as an array of numbers is a shorthand

too.

This is what Matrices really are: they are linear transformations.

Ok so let me explain. Remember vectors from last time? Well imagine you have a some vector

and want to transform it into some other vector instead. In fact say you have some whole vector

space, let’s say all the vectors in 2D space, and you want to transform them into some vectors

in another vector space or the same space, for example vectors in 3D space. Then a transformation

is just something that eats a vector from your initial space and spits out a vector

in your new space.

A Matrix is a transform. But a very specific type called a linear transform, which is something

that respects linear combinations. Let’s say that I have two vectors. I’m going to

apply this matrix to them each and that will transform them in some way. Then, I can make

a linear combination of the result. But what if I decided to do this linear combination

first, and then put the resulting vector into the matrix.

A linear transformation is one where these to processes result in the same final vector.

In other words, you can first put your two vectors into the matrix, and then take a particular

linear combination of them or you can take the linear combination of them first, then

apply the matrix. Either way you will get the same result.

To see if you’ve understood, I want you to figure out if the following transformations

are linear transformations or not. The first is a transformation that takes a

2d vector and then just multiplies its length by 3.

The second is like this. First draw a line through the origin. Then take any vector in

2d, say this one, and draw a perpendicular from that line to the tip of the vector. The

new vector is on the line.

So are these two linear transformations? Think about the answer and then put it in the poll

in the corner. Remember that committing to an answer like this helps you learn.

Ok, so the answer might surprise you but the answer is that they’re both linear. To see

this for the first one, take any two vectors and apply the transformation to them then

make a linear combination of them. This is clearly the same as if you made the linear

combination and then multiplied the result by 3. The second one is a bit tricker, but

the easiest way to see it is, first draw two vectors. Then notice that these vectors are

equal to these two perpendicular vectors added together.

When you apply the transform to them, what it does is it cuts off this bit. Then if you

add them you get this. But if you’d instead added the original vectors first, it would

be like adding all these 4 bits first, which you can group up like this. Now if we apply

the transform it cuts off this bit, and see, we have the same thing in both cases. Kind

of neat right?

But here’s a really nice thing about linear transforms. Imagine you have a linear transform

from some initial space, to some other space and you have an arbitrary basis for your initial

space Now let’s suppose that we know exactly what

the linear transform does to the basis vectors. This is in fact enough to figure out what

it does to the entire space. To see why, say you have some other vector in this initial

space. You can write it as a linear combo of basis vectors- in 2d remember you do this

by using

a grid. Ok, once you’ve finally got that linear combination if you apply the linear

transformation to it, you can see by the linearity that you only need to know what happens to

the basis to compute this thing.

So now we get to writing a matrix as an array. What you’ll see is, this first column represents

what happens to your first basis vector, the second what happens to your second etc. Remember

how to write vectors as columns in some basis? It’s a shorthand for writing out this linear

combination of basis vectors. Well then clearly the basis vectors are written like this.

Say I apply my linear transform to the first basis element and I get this column vector,

and I do this for each basis vector. As we said before this is everything I need to know

about this linear transform, so I want to store this information compactly, so I store

it in a matrix like this. And that is really all a matrix is. It’s shorthand, where,

just like in the case of vectors written as columns, the basis is hidden. But the basis

is super important, so this array of numbers only represents a linear transformation as

long as you’re clear what the basis is.

Now I can explain what it means to ‘multiply’ a matrix by a vector. It’s not about multiplication

at all. All it means is you apply the linear transformation that this matrix represents

to the vector this represents. As we saw, this is just equal to this big sum.. Which

you might recognise as the rule for multiplying matrices and vectors.

To see if you’ve got this, see if you can come up with a matrix for the following linear

transformation: It takes the first basis vector and does nothing to it, and takes the second

basis vector and deletes it. Another way of thinking of that is, it takes the second basis

vector to the zero vector- the vector with no length. Pause the video now and try to

come up with the matrix in this basis. Hopefully you’ve had a think. The answer is this.

By the way, your matrices definitely don’t need to be square. The dimensions of the matrix

tell you about the dimension of the vectors it eats vs the dimension of the vectors it

spits out. In this example, this matrix takes 3d vectors,

and gives back 2d ones. Then it’s shape has to be this, because it has 3 rows, representing

what the matrix does to the 3 basis vectors of the initial space, and 2 columns because

each basis vector becomes a 2d vector.

If you’ve following everything so far, then matrix multiplication is going to be really

simple to explain. You know how when you first meet matrix multiplication, they throw this

crazy formula at you. Then they expect you to act shocked when you find out, if you call

this insane thing multiplication, then A times B doesn’t equal B times A. But there is

a much much easier way to understand matrix multiplication. Let’s look at an example

first.

Imagine if you want to transform some vectors. First you want to rotate them by 90 degrees

in the plane. Then you want to project them onto this line like we did earlier. Both of

these are linear, so call the matrix representing these transformations in some basis A and

B. Here’s a question. Is it the same thing to do the projection first, then the rotation?

I can tell you, it’s not, but I want you to pause the video now and think of a vector

that would come out different if you do the projection first or if you do it second. If

you could think of one, say so in the poll here.

Ok, so there are many options but here’s a vector that works: If the vector starts

on the line, and you apply A it gets rotated to here, then when you apply B to the result

you project it down, but in this case, nothing happens to it. But if you did it in the other

order when you apply B it immediately becomes the zero vector, and then when you apply A

nothing else happens. So clearly doing A then B is different to doing B then A

So here’s the big secret: When we write B A, the linear transformation that we’re

talking about is the one where you do A first and then B. That’s all it means. And as

we’ve just seen, order matters, so this is different from the linear transformation

A B, since in that case you swap the order. So the statement AB doesn’t equal BA is

actually blindingly obvious, once you understand the meaning of these things and really shouldn’t

be presented as a startling fact.

So why does this lovely idea of composing maps lead to the awfulness that is the matrix

multiplication formula? Well actually, once you understand it, the formula isn’t really

ugly at all. Let me show you what I means for the case when A and B are 2x2 matrices.

How would we try and write AB as a matrix in a particular basis? Well, let’s follow

our own rules. We need to find out what AB does to the each of the basis vectors. Let’s

do the first one. Well, when we apply B, we get this first column of B, by definition.

But then when we apply A, we get b_11 times the first column of A plus b_21 times the

second column. This result is our first column of AB. Do the same for the second basis vector,

and here is the formula. See? Matrices aren’t so painful, after all.

OK, It’s homework time!

First, Not all sized matrices can be multiplied together. Think about it in terms of them

representing transformations from one space to another, and figure out which size matrices

can be multiplied and explain why in the comments.

Second, and this is a multiple choice one, consider a transformation that takes a 3d

vector, and adds some fixed vector k to it. Say k is the vector (7 3 3). Is this a linear

transformation or not?

Third, imagine you have a matrix A that multiplies the first basis vector by 2, and the second

basis vector by 6. How do you write A in this basis?

As you will have noticed, these questions are from Brilliant.org who are sponsoring

this video. I said this last time, but I think linear algebra is just a topic you can’t

understand without doing examples. If you try and understand the topic just through

learning a bunch of formulas or theorems you just won’t be able to get all the subtleties

involved. I found that when I went through Brilliant’s course, which has really well

designed examples, I remembered a bunch of little tricky points that I’d forgotten

about, and it got me really comfortable with doing actually doing calculations again- since

I don’t do a lot of that anymore.. You can sign up to either a monthly or yearly membership

and access this course and many others on their website. If you follow the link on the

screen or in the description you can get 20% off the yearly membership.

Thanks for watching these linear algebra videos, it’s one of my favourite fields of maths

so I love teaching it because it feels like I’m learning all again myself. Anyway, the

next one will be linked here in 2 weeks so subscribe if you’d like to see it when it’s

out.