This video is sponsored by Brilliant.org
This is the start of a mini series about linear algebra, the study of vectors and matrices.
I’m making these videos for 2 reasons.
The first is that linear algebra is the language of quantum mechanics so without some of this
background I couldn't eventually teach you some topics in my quantum series, like quantum
computing for example.
The second is, this topic is so geometrical and beautiful and yet, the way it’s sometimes
taught in university and high school, students don't take away a lot of intuition about it.
I’m hoping that these videos might show you linear algebra from a slightly different
perspective than you might have seen before.
This is the perspective that first made it all click for me, so I hope it might work
for some of you too.
If not, don’t worry, at the end of the video I’ll point you to some other resources I
really like that might help instead.
Anyway, let’s start with vectors.
We’ll see later lots of different things count as vectors, but for now we’re going
to think of them very concretely.
Vectors are arrows, with some fixed direction, and some length.
I’m going to tell you a way of thinking of vectors that sounds really silly, but that
I found very helpful.
Let’s think of them as arrows that tell you where to go.
A vector says, if you start here, you need to go this much, in this direction.
Basically, if you put the dull side of the vector where you start, you end up on the
pointy side.
Thinking of vectors this way makes it really easy to add vectors.
When you’re introduced to vectors, you might think, if a vector is 3 units long, and another
vector is 2 units long, if you add them, you get something that’s 5 units long.
But that’s not necessarily true, and you can see why if you think of it this way.
When you add these two vectors, what you’re really saying is, if I go the direction the
first vector told me to go, then from that end point go the direction the second vector
wanted me to go, in total how far did I go from the start, and in what direction?
In this example, the vector representing your overall path is this, and so that’s the
sum of these vectors.
By the way, notice that if you’d followed this vector first, then this one, you’d
end up in the same spot, so the order doesn’t matter at all.
But something that does matter is the direction.
You can’t rotate these vectors around when you add them.
To see if you understood all this so far, here’s a quick question.
Imagine you add these two vectors.
What’s the sum of them?
Here are your options.
Pause the video to think about it, and when you’re ready you can vote for your answer
in the poll here.
To answer this, you need to move this vector over here.
But.
If you put the vector this way, then you would be following this vector in the opposite direction
than it wants you to go.
Instead, you need the vector like this, and you end up going almost back the way you came,
but not quite.
So this is your resulting vector.
As you can see here, another way to add vectors then is by drawing these parallelograms- but
this is just a useful shortcut.
Anyway, now that we understand how to add 2 vectors, we understand how to add as many
as we like.
Just.
keep.
following.
directions.
Good.
But adding together isn’t all ve ctors can do.
They can also get multiplied by a number.
There’s a simple way to think of this too.
This just stretches or squeezes the length of the vector, without changing the direction.
But what happens when you multiply it by negative one for example?
It just flips the vector in the opposite direction.
And if you have negative a third for example?
That just flips it’s direction, then squishes it by a 1/3.
Before we move on to the really interesting stuff, a quick word about notation.
I’ll write a general vector like this, with a little arrow on top.
This is fairly common notation, but there are lots different ways people use.
In fact, in quantum mechanics we use a notation you’ve already seen if you watch my videos,
and it’s like this.
Anyway, say you have some vectors.
Now you know you can add them together, and you can also multiply each by a number that
stretches it or squishes it.
But the most general thing you can do is to do both.
Multiply each of the vectors by something, and then add them together.
This is called a linear combination of the vectors, and it’s the most general way that
you can make new vectors from ones you have.
A question you might be asking yourself is, exactly which other vectors can I make with
a given set of vectors.
For example, if I give you these two vectors, can you end up with any vector on plane by
taking linear combinations?
How about see if you could make this vector?
On a piece of paper, try drawing something like this, doesn’t have to be exactly the
same, and see if by stretch squeezing and flipping the original vectors, then adding
them, you can make it this third vector.
I think the best way to understand vectors is to actually play around with them, so seriously,
pause the video, pick up a scrap of paper and try it.
And if you think you know how to do, try it with another example as well.
Assuming you gave that a go, here’s how you do it for any example.
Draw a grid like this with your two vectors.
This says, you need to go 2 of these lengths to get to here, and minus 1.5 to get to here.
So in my case, the new vector is: -1.5 v_1+2 v_2
Using this method, you can take a linear combination of 2 vectors get any other vector on this
plane.
Well not quite.
There’s one special case.
Imagine if I had two vectors that are on the same line.
One of them is just a multiple of the other.
So when you do a linear combo of them, you’re really just getting a multiple of one of them.
The other one is redundant.
If we look at all the vectors you can make from these vectors, which is called the span
of the vectors, it’s just anything on this line.
But let’s see what happens when we think of the 3D case.
How many vectors do you need so that a linear combinations of them will give you every possible
vector in 3d space?
Clearly two vectors is not enough.
Any 2 vectors only get you vectors on the plane they span.
So you need a third vector.
But what if that third vector is on the same plane?
Then clearly it’s redundant.
Another way to think of this is, you can write the third vector as a combination of the other
two.
So when you have a linear combination of all three vectors, you can just rewrite it as
a linear combination of just the two vectors instead.
So the third vector didn’t help at all.
On the other hand, if your third vector is not in the plane, you can get to any vector
anywhere in 3D.
You can convince yourself of that by using a similar grid argument to the 2D case.
If you’ve understood everything so far, you’ve understood one of the most important
ideas in linear algebra, and that is the idea of a basis.
We saw in our examples that a line can be spanned by just one vector, a plane can be
spanned by 2 and 3D space can be spanned by 3 vectors.
But only if you don’t have any redundancies.
Imagine that you have vectors v_1 to v_n spanning some space.
A vector, say v_i, is redundant if, when you take that vector out, you still get the same
space.
That happens when that vector can be written as a linear combination of the other vectors
in the set.
As long as none of the vectors are dependent on the other vectors like this, we have the
minimum number of vectors necessary to span the space- if we deleted any, we’d get a
smaller space.
Then this set of vectors is called the basis for the space.
Obviously, for any space of vectors, there are many different choices of basis for example
these are both bases for the plane, so it’s not unique.
But here are 2 reasons why they’re still very important.
Firstly, for any vector space, the number of basis elements is fixed.
It’s called the dimension of that space.
The second thing is, if you pick a basis, and you write another vector in that space
as a linear combination of these basis elements, there’s only one correct way to do it.
I.e, you can’t write that v is also equal to this other linear combination.
It’s not hard to prove these 2 statements and I think it’s really good practice for
you, so I’d like you to have a go at proving them and write your solution in the comments.
Now, finally, I want to explain the link between what I’ve said so far and what you might
have seen about vectors before.
You may have seen vectors written as a column of numbers like this.
I think a lot of students misunderstand what this means.
A vector isn’t really a column of numbers; this is really a short hand.
What it means is that, there’s some basis that you have for your vectors, fixed already.
So when you write your vector as a column you’re really saying, this vector is a times
the first basis vector plus b times the second basis vector etc.
It’s just a short hand for writing out this whole sum, and it’s very convenient.
But, I really hope, that if I taught you nothing else, vectors are not a stack of numbers.
The are arrows with directions, and you can only write them as a column of numbers once
you’ve established what the basis is.
Before you go, I’d really encourage you to try the following 2 multiple choice questions.
I strongly believe that linear algebra is one of those topics you can’t understand
without getting your hands dirty.
Even if you think everything in this video was very straight forward, there are lots
of tricky questions that can help you understand the subtleties.
Here are 2 that, on the surface seem quite simple, and the calculation would only take
you a minute, but you need to think a bit carefully.
So here are the questions.
You can find both of them in the description as well.
The first question is about redundancy.
The question is, one of these vectors is redundant.
Which one is it?
Once you’ve had a think, put your answer in the poll here.
The second question is about which of these vector spaces contains the others.
Again, you can put your answer in the poll in the corner here.
I’d be really interested to what you guys thought.
As you may have noticed, these two questions are from Brilliant.org, who are sponsoring
this series.
Brilliant have a course on linear algebra which I think will complement this videos
perfectly because they introduce the subject from a different angle, by looking at systems
of linear equations.
I’ve never seen that approach done as well as their website does it, and I think the
reason it works so well is that they don’t just tell you the concepts- they ask you questions
which make you engage with the material.
If you watch my channel, you know that I set homework at the end of my videos for exactly
the same reason -you can’t learn without thinking about the material on your own.
The questions at Brilliant are at the perfect level to keep you on your toes and to illustrate
the concepts in a very tangible way.
Despite the fact that I do research in field that is basically applied linear algebra,
I got a lot from going through their course, so I can personally recommend it.
I really wanted all of you to have the chance to do the part of their linear algebra course
relevant to this video, so they've kindly made that free for the next two weeks.
If you like their stuff, you can get 20% off their annual membership by following the link
in the description.
My next video will be up here soon, where we’ll learn about Matrices.
Meanwhile, I really recommend a video series by 3blue1Brown on linear algebra, especially
if you’d like to go deeper in this topic.