This week's topic is Linear Algebra.
It'll actually be the topic for the next
two weeks.
And the focus of this course is going to
shift a little bit.
So the first half, the first four weeks of
lectures, I was basically aiming at doing
a calculus review.
And, getting everybody up to the level
where they were able
to Take the partial derivatives of the
black scholes pricing formulas.
Now what I'm sort of driving at, is a
problem from portfolio theory.
And so we're going to use tools from
linear algebra to describe a portfolio.
And then in the, in the third week.
We'll look at some ways some ways to do
mathematical optimization.
So for a given sort of target return
of your portfolio you'll want to minimize
the risk.
And then in the final week we'll look
at some numerical methods for solving
problems like that.
So this week's outline today I'm going to
try and cover, topics one through five.
So, vectors, this is just points, so a
vector of length
two, you could think of that as a point in
a plane.
Vector of length three, you could think of
that as a point
with x, y, z coordinates and then you can
just keep going.
So you can have a vector in sort of five
dimensional space.
That's just
a vector with five numbers in it.
[COUGH]
In section two, I'll talk about vector
length
and then plane, so that's two separate
topics.
But they're kind of short, so I mushed
them into one video.
Then, we'll look at,
Solving systems of linear equations.
So, if I have a, three equations in three
unknowns, basically, and the way
you would normally think about this, each
one of those equations determines a plane.
Two planes will intersect in a line.
And then the third plane will intersect
that line in a single
point, and that'll be the solution to my
system of linear equations.
And then we'll start looking at some
concepts from
linear algebra that allow us to solve that
problem,
so something called elimination.
And then I'm going to use elimination
to motivate the idea of matrix
multiplication.
Then, next week we'll look at actually
solving,
so Ax equals b, so this is just the.
Matrix form of a system of linear
equations.
And we're going to look at solving that.
And sort of the moral of this set up
lectures is that
we can solve problems of this form Ax
equals b or
in general a lot, any problem coming up in
linear algebra.
Usually by factoring a matrix so there are
different ways that a matrix can be
factors.
So this is breaking it into a product of,
so a
set of matrices whose product is equal to
the original matrix.
And then hopefully that representation of
the matrix is going to
make it easier to solve the problem we're
trying to do.
And then I'm going to finish by
introducing
something called the R Environment for
Statistical
Computing, and this is a software
environment
that lets you manipulate vectors in
matrices.
And so I think a lot of these topics will
be become
a lot more clear once you actually try to
use them to
solve a particular problem, and that you
can do that with a
pen and paper but it ends up being a lot
of work.
And so I don't really want to stress doing
that.
I want to stress
understanding the concepts, and then
solving them using a computer.
The first topic will be vectors.
So imagine we have a, a portfolio.
So a portfolio's just a, a collection of
assets.
literally a portfolio just means a folder,
so it's, something like this
that I use to collect your quizzes at the
end of the day.
And I suppose in the old days, people used
to put their shares
into a portfolio, and so that, the word is
sort of stuck around.
And I'm going to consider a very simple
portfolio.
So I have two assets, which I cleverly
named asset 1 And asset 2.
And then I'll have w1 shares of asset 1,
and w2 shares of asset 2.
So, you could think of this as me, say
owning five shares of Apple and three
shares of Google.
Something like that.
But, instead of just kind of making a
list, I want to
think of this as a, something called a two
dimensional vector.
So I'm going to call my vector w, and
it'll have components w1 and w2.
And so, in general you want to think of a
vector
because our goal is to eventually combine
this with matrices.
So there's two ways you can write a
vector.
I could write it as a row, which is sort
of this way, or
I can write it as a column, which is this
way that's in black.
And generally, I want to think of vectors
as always being a column,
but sometimes just because it's
inconvenient to
write them, you know if I have a
vector of ten things it's going to take up
the whole slide just to write one line.
So I'll write them in a row, but I'll use
parentheses instead.
So, this means the same thing.
If it's got the square brackets, I mean
the vector
actually the way it's written down, and if
it's got the
round brackets, parentheses, then even if
I write it like
this, what I really mean is a column with
w1 on
top and w2 on the bottom.
And w1 and w2 are called the components of
the vector, w.
And so the you know simplest operation
would be addition.
So, suppose I had a second portfolio.
So I have u shares of asset 1 and u shares
of asset 2.
So I can have different numbers but the
interpretation has to be the same.
So Whatever number goes in the first
component of
my vector, that's the number of shares of
asset 1.
Whatever number goes in the second
component of my
vector, that's the number of shares of
asset 2.
And then,
you know, this is just two separate
portfolios.
If I were to put them together, it should
be pretty clear that it's going to be
the, the number of shares of each asset
that are going to be in, the combined
portfolio.
So I'm going to define addition just to
work like that.
And this isn't just addition for vectors
that
represent portfolios, this is, this is
addition in
general for vectors and I'm just trying to
use this idea of a portfolio of assets
to, to help you understand.
So if I have a vector u and a vector w,
and I add them together.
I'm just going to add them component-wise,
so the first component of the sum
is equal to the sum of the two first
components, and so on.
And so the, I think I already set it, but
the.
So to the language I want to use is
component-wise.
So if I do an operation on a vector that
effects each
component of the vector individually, I'll
use the, the word component wise.
And any.
Suppose I want to, consider the problem of
multiplying a vector by a
scalar quantity, so scalar quantity is
just going to be a real number.
But it's a single real number, so vector
has more than one component, a scalar is
just one single real number so if I looked
at just multiplying
a vector by an integer, and so, 2 is a
pretty easy integer to work with.
Normally,
I would think of 2 times w as just w times
w, and so then
if I did that using my addition rule, I
would just end up with 2w1 2w2 as the sum.
And so in general I'm going to do the same
thing.
So, this scaler here that I put in front
of the
vector, just ends up getting multiplied
into each component of the vector.
So when I define multiplying a vector by a
scaler so
here, this value c is a scaler value, some
real number.
Then the answer is just going to be c
times the first component and c times the
second component.
Then I can make something called a linear
combination of two vectors.
So again I have a vector u and a vector w.
And
I, I now have two scalar quantities, c1,
ops, and c2.
And I'm just going to combine my scalar
rule and my addition
rule.
So c1 times u becomes cu1, c1u1,
c1u2.
c2w becomes c2w1 c2w2, and then I just add
those component wise, and so I end up with
this
as the expression for the first component
of this linear
combination, and this as the expression
for the second component.
And then one nice thing I can get out of
here is if I set c1 equal to 1,
ops, and c2 equal to minus 1.
Then I end
up with u minus w so this allows me to
define Vector subtraction.
And I guess I should, should mention I'm,
I'm using this vectors of length too just
because I want to have something to show
off that's in the form of a vector.
This works for vectors of any length just
again if they get longer it's just that
much more type setting for me and not much
less that will fit on the slides.
And so in mathematics generally people
tend to trust
something more if they can draw a picture
of it.
And so we can actually draw a picture of
vector addition.
So suppose a vector W So, again it's just.
I'm going to use points and the planes.
So, the vector w will have two components.
And, we'll draw that with an arrow.
And, I'll put the tail
of the arrow at the origin.
So, for any vector, the tail always goes
at the origin.
And then the pointy end goes at the point
w1, w2.
So it goes at the point whose components
are the components of the vector.
So for example if I let u equal the vector
1,2 and w equal the vector
3,1 then I end up with a picture like
this.
So, to get u, I put the tail of the
arrow, so that's the not pointy end at the
origin.
I put the pointy end at the point 1,2,
so this is the x coordinate, the first
component,
if you're describing points on the plane
you're
going to think of that as the x component.
And the second one will be the y
component.
So I just put the pointy end at 1, 2.
Same for w, I put the x at 3, and y is 1.
So the pointy end goes at the point 1, 3.
And then if I add these together, what
ends up
happening is I can think of addition as
taking one vector,
so for this bottom path to the sum, I can
think of that as taking this
vector u, and drawing its tail at the tip
of w.
So I go, w like that, and then I go u like
that, and
then I end up at the vector v, which is
this blue arrow here.
And I could have done it the other way.
That's a, it's an associative, I'm sorry,
commutative operation so I could have
taken,
I could have gone from here to here so
I've gone up to u.
And then taken the vector, w, and placed
it here,
and that would also get me to the vector,
v.
And then subtraction.
So if I start out with v, and I
want to find out what is v minus u, so I'm
aiming to do this operation right here.
I'll take the vector v, then minus 1 times
u.
So if I multiply a vector by a
scalar, that's just changing the length of
the arrow.
But it's not changing the direction that
it's pointing in.
So if I multiply it by negative one, I'm
just taking this vector, and, ops.
well, here, here's u.
It just ends up being this u, but pointing
the other direction.
So I would draw that from
the origin, down.
But since I'm doing an addition operation,
I move the origin from my vector, u.
To be this vector v here.
So to get w, I follow v, and then I follow
u, and I end up at w.
So, since each component in the sum So
here I'm using i to mean the components,
so
in my case I've been looking at vectors of
length two so I could be one or two.
So, since ui plus wi is equal to wi plus
ui because addition is commutative.
I have the same property for vectors.
So u plus w is equal to w plus u.
So that should be pretty obvious.
I just have 1 plus 3 is 4 or 3 plus 1 is 4
and so on.
There is one special vector called the
zero vector.
And that has each component equal to zero.
So every component in the vector has to be
zero.
And then it's going to have the same
property that zero
has for regular numbers if I add zero to
any vector.
So you have to be a little bit careful
here,
because the zero vector has to be the same
length.
As the vector w,
but generally if w is a vector and I add 0
to
it, you just sort of have to look at that
and say, well,
this 0, the type of 0 has to match the
type of
this w in order for me to be able to do
this operation.
So, 0 must be a vector of 0, the same
length as w.
And the zero vector has the same property
of zero, so if
I add it to any vector, I just get my
vector back.
If I scale a vector, so here I've used 2,
but in general this could be any real
number c.
this pres, well To preserve the direction,
it can be any real number except for zero.
And, if I put two here, it's going to
double the length.
If I put an arbitrary constant c here, it
will be c times the length.
And then minus u, you can think of that as
being the same length as u.
By just pointing in the opposite
direction.
So that sort of covers addition and then
these, the
operations of dealing working with vectors
using scalars.
We can also Introducing a concept
called the dot product or inner product.
And this gives us some notion of how you
could multiply two vectors together.
So if I have two vectors u and w that
are length two, the dot product is just
going to be the scalar quantity.
So here I'm, I'd read this left side as u
dot w.
And it's just defined to be
the product of the first component.
The product of the first component plus
the product of the second components.
Or in general if this a, longer vector.
So w or length three.
Or length four.
Or length five.
It's the, the product of the, the
components with
the same index, and then all of those
sumed together.
So if it was length four, it would be u1w1
plus u2w2 plus u3w3 plus u4w4, and
then, all summed up.
And so this going to always be a real
number.
And just a, a scale or quantity.
It's not goig to be a vector.
So just a, a simple example.
If I take the dot product of 1, 2 and 3,1
I end up with 1 times 3 plus 2 times 1 so
1 times 3.
Ops, plus 2 times 1, and that'll be equal
to 5.
And the reason this type of product makes
sense, so again,
this is something we can actually
understand again, with my portfolio
example.
So remember, w was my position in assets 1
and 2.
So w1 was how many shares of asset 1 I
own.
And w2 was how many shares of asset 2 I
own.
And suppose I have another vector p1 and
p2
that are the prices of asset 1 and asset
2.
Then if I define v to be the dot product
of.
W and p so w dot p.
That's going to be the number of shares of
asset
1 times the price of asset 1 plus the
number
of shares in asset 2 times the price of
asset
2 and that'll just be the value of my
portfolio.