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This week's topic is Linear Algebra.

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It'll actually be the topic for the next
two weeks.

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And the focus of this course is going to
shift a little bit.

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So the first half, the first four weeks of

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lectures, I was basically aiming at doing
a calculus review.

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And, getting everybody up to the level
where they were able

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to Take the partial derivatives of the
black scholes pricing formulas.

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Now what I'm sort of driving at, is a
problem from portfolio theory.

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And so we're going to use tools from
linear algebra to describe a portfolio.

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And then in the, in the third week.

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We'll look at some ways some ways to do
mathematical optimization.

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So for a given sort of target return

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of your portfolio you'll want to minimize
the risk.

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And then in the final week we'll look

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at some numerical methods for solving
problems like that.

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So this week's outline today I'm going to
try and cover, topics one through five.

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So, vectors, this is just points, so a
vector of length

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two, you could think of that as a point in
a plane.

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Vector of length three, you could think of
that as a point

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with x, y, z coordinates and then you can
just keep going.

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So you can have a vector in sort of five
dimensional space.

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That's just

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a vector with five numbers in it.

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[COUGH]

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In section two, I'll talk about vector
length

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and then plane, so that's two separate
topics.

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But they're kind of short, so I mushed
them into one video.

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Then, we'll look at,

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Solving systems of linear equations.

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So, if I have a, three equations in three
unknowns, basically, and the way

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you would normally think about this, each
one of those equations determines a plane.

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Two planes will intersect in a line.

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And then the third plane will intersect
that line in a single

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point, and that'll be the solution to my
system of linear equations.

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And then we'll start looking at some
concepts from

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linear algebra that allow us to solve that
problem,

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so something called elimination.

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And then I'm going to use elimination

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to motivate the idea of matrix
multiplication.

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Then, next week we'll look at actually
solving,

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so Ax equals b, so this is just the.

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Matrix form of a system of linear
equations.

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And we're going to look at solving that.

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And sort of the moral of this set up
lectures is that

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we can solve problems of this form Ax
equals b or

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in general a lot, any problem coming up in
linear algebra.

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Usually by factoring a matrix so there are

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different ways that a matrix can be
factors.

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So this is breaking it into a product of,
so a

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set of matrices whose product is equal to
the original matrix.

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And then hopefully that representation of
the matrix is going to

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make it easier to solve the problem we're
trying to do.

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And then I'm going to finish by
introducing

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something called the R Environment for
Statistical

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Computing, and this is a software
environment

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that lets you manipulate vectors in
matrices.

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And so I think a lot of these topics will
be become

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a lot more clear once you actually try to
use them to

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solve a particular problem, and that you
can do that with a

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pen and paper but it ends up being a lot
of work.

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And so I don't really want to stress doing
that.

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I want to stress

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understanding the concepts, and then
solving them using a computer.

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The first topic will be vectors.

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So imagine we have a, a portfolio.

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So a portfolio's just a, a collection of
assets.

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literally a portfolio just means a folder,
so it's, something like this

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that I use to collect your quizzes at the
end of the day.

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And I suppose in the old days, people used
to put their shares

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into a portfolio, and so that, the word is
sort of stuck around.

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And I'm going to consider a very simple
portfolio.

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So I have two assets, which I cleverly

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named asset 1 And asset 2.

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And then I'll have w1 shares of asset 1,
and w2 shares of asset 2.

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So, you could think of this as me, say

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owning five shares of Apple and three
shares of Google.

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Something like that.

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But, instead of just kind of making a
list, I want to

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think of this as a, something called a two
dimensional vector.

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So I'm going to call my vector w, and
it'll have components w1 and w2.

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And so, in general you want to think of a
vector

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because our goal is to eventually combine
this with matrices.

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So there's two ways you can write a
vector.

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I could write it as a row, which is sort
of this way, or

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I can write it as a column, which is this
way that's in black.

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And generally, I want to think of vectors
as always being a column,

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but sometimes just because it's
inconvenient to

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write them, you know if I have a

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vector of ten things it's going to take up
the whole slide just to write one line.

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So I'll write them in a row, but I'll use
parentheses instead.

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So, this means the same thing.

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If it's got the square brackets, I mean
the vector

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actually the way it's written down, and if
it's got the

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round brackets, parentheses, then even if
I write it like

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this, what I really mean is a column with
w1 on

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top and w2 on the bottom.

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And w1 and w2 are called the components of
the vector, w.

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And so the you know simplest operation
would be addition.

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So, suppose I had a second portfolio.

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So I have u shares of asset 1 and u shares
of asset 2.

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So I can have different numbers but the
interpretation has to be the same.

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So Whatever number goes in the first
component of

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my vector, that's the number of shares of
asset 1.

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Whatever number goes in the second
component of my

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vector, that's the number of shares of
asset 2.

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And then,

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you know, this is just two separate
portfolios.

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If I were to put them together, it should
be pretty clear that it's going to be

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the, the number of shares of each asset

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that are going to be in, the combined
portfolio.

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So I'm going to define addition just to
work like that.

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And this isn't just addition for vectors
that

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represent portfolios, this is, this is
addition in

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general for vectors and I'm just trying to
use this idea of a portfolio of assets

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to, to help you understand.

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So if I have a vector u and a vector w,
and I add them together.

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I'm just going to add them component-wise,
so the first component of the sum

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is equal to the sum of the two first
components, and so on.

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And so the, I think I already set it, but
the.

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So to the language I want to use is
component-wise.

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So if I do an operation on a vector that
effects each

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component of the vector individually, I'll
use the, the word component wise.

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And any.

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Suppose I want to, consider the problem of
multiplying a vector by a

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scalar quantity, so scalar quantity is
just going to be a real number.

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But it's a single real number, so vector
has more than one component, a scalar is

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just one single real number so if I looked
at just multiplying

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a vector by an integer, and so, 2 is a
pretty easy integer to work with.

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Normally,

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I would think of 2 times w as just w times
w, and so then

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if I did that using my addition rule, I
would just end up with 2w1 2w2 as the sum.

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And so in general I'm going to do the same
thing.

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So, this scaler here that I put in front
of the

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vector, just ends up getting multiplied
into each component of the vector.

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So when I define multiplying a vector by a
scaler so

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here, this value c is a scaler value, some
real number.

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Then the answer is just going to be c

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times the first component and c times the
second component.

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Then I can make something called a linear
combination of two vectors.

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So again I have a vector u and a vector w.
And

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I, I now have two scalar quantities, c1,
ops, and c2.

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And I'm just going to combine my scalar
rule and my addition

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rule.
So c1 times u becomes cu1, c1u1,

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c1u2.

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c2w becomes c2w1 c2w2, and then I just add

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those component wise, and so I end up with
this

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as the expression for the first component
of this linear

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combination, and this as the expression
for the second component.

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And then one nice thing I can get out of
here is if I set c1 equal to 1,

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ops, and c2 equal to minus 1.
Then I end

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up with u minus w so this allows me to
define Vector subtraction.

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And I guess I should, should mention I'm,
I'm using this vectors of length too just

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because I want to have something to show
off that's in the form of a vector.

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This works for vectors of any length just
again if they get longer it's just that

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much more type setting for me and not much
less that will fit on the slides.

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And so in mathematics generally people
tend to trust

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something more if they can draw a picture
of it.

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And so we can actually draw a picture of
vector addition.

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So suppose a vector W So, again it's just.
I'm going to use points and the planes.

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So, the vector w will have two components.
And, we'll draw that with an arrow.

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And, I'll put the tail

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of the arrow at the origin.

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So, for any vector, the tail always goes
at the origin.

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And then the pointy end goes at the point
w1, w2.

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So it goes at the point whose components
are the components of the vector.

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So for example if I let u equal the vector
1,2 and w equal the vector

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3,1 then I end up with a picture like
this.

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So, to get u, I put the tail of the

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arrow, so that's the not pointy end at the
origin.

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I put the pointy end at the point 1,2,

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so this is the x coordinate, the first
component,

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if you're describing points on the plane
you're

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going to think of that as the x component.

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And the second one will be the y
component.

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So I just put the pointy end at 1, 2.

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Same for w, I put the x at 3, and y is 1.
So the pointy end goes at the point 1, 3.

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And then if I add these together, what
ends up

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happening is I can think of addition as
taking one vector,

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so for this bottom path to the sum, I can
think of that as taking this

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vector u, and drawing its tail at the tip
of w.

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So I go, w like that, and then I go u like
that, and

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then I end up at the vector v, which is
this blue arrow here.

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And I could have done it the other way.

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That's a, it's an associative, I'm sorry,

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commutative operation so I could have
taken,

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I could have gone from here to here so
I've gone up to u.

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And then taken the vector, w, and placed
it here,

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and that would also get me to the vector,
v.

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And then subtraction.
So if I start out with v, and I

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want to find out what is v minus u, so I'm
aiming to do this operation right here.

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I'll take the vector v, then minus 1 times
u.

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So if I multiply a vector by a

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scalar, that's just changing the length of
the arrow.

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But it's not changing the direction that
it's pointing in.

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So if I multiply it by negative one, I'm
just taking this vector, and, ops.

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well, here, here's u.

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It just ends up being this u, but pointing
the other direction.

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So I would draw that from

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the origin, down.
But since I'm doing an addition operation,

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I move the origin from my vector, u.
To be this vector v here.

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So to get w, I follow v, and then I follow
u, and I end up at w.

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So, since each component in the sum So

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here I'm using i to mean the components,
so

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in my case I've been looking at vectors of
length two so I could be one or two.

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So, since ui plus wi is equal to wi plus
ui because addition is commutative.

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I have the same property for vectors.
So u plus w is equal to w plus u.

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So that should be pretty obvious.

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I just have 1 plus 3 is 4 or 3 plus 1 is 4
and so on.

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There is one special vector called the
zero vector.

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And that has each component equal to zero.

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So every component in the vector has to be
zero.

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And then it's going to have the same
property that zero

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has for regular numbers if I add zero to
any vector.

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So you have to be a little bit careful
here,

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because the zero vector has to be the same
length.

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As the vector w,

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but generally if w is a vector and I add 0
to

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it, you just sort of have to look at that
and say, well,

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this 0, the type of 0 has to match the
type of

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this w in order for me to be able to do
this operation.

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So, 0 must be a vector of 0, the same
length as w.

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And the zero vector has the same property
of zero, so if

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I add it to any vector, I just get my
vector back.

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If I scale a vector, so here I've used 2,

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but in general this could be any real
number c.

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this pres, well To preserve the direction,
it can be any real number except for zero.

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And, if I put two here, it's going to
double the length.

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If I put an arbitrary constant c here, it
will be c times the length.

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And then minus u, you can think of that as
being the same length as u.

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By just pointing in the opposite
direction.

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So that sort of covers addition and then
these, the

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operations of dealing working with vectors
using scalars.

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We can also Introducing a concept

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00:15:30,410 --> 00:15:34,140
called the dot product or inner product.
And this gives us some notion of how you

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could multiply two vectors together.
So if I have two vectors u and w that

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are length two, the dot product is just
going to be the scalar quantity.

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So here I'm, I'd read this left side as u
dot w.

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And it's just defined to be

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00:15:58,290 --> 00:15:59,854
the product of the first component.

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The product of the first component plus
the product of the second components.

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Or in general if this a, longer vector.
So w or length three.

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Or length four.
Or length five.

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00:16:09,730 --> 00:16:13,270
It's the, the product of the, the
components with

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00:16:13,270 --> 00:16:16,290
the same index, and then all of those
sumed together.

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00:16:16,290 --> 00:16:24,244
So if it was length four, it would be u1w1
plus u2w2 plus u3w3 plus u4w4, and

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then, all summed up.

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And so this going to always be a real
number.

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00:16:31,630 --> 00:16:34,380
And just a, a scale or quantity.
It's not goig to be a vector.

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So just a, a simple example.
If I take the dot product of 1, 2 and 3,1

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I end up with 1 times 3 plus 2 times 1 so
1 times 3.

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Ops, plus 2 times 1, and that'll be equal
to 5.

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And the reason this type of product makes
sense, so again,

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00:16:57,530 --> 00:16:58,840
this is something we can actually

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00:16:58,840 --> 00:17:01,830
understand again, with my portfolio
example.

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00:17:03,520 --> 00:17:06,910
So remember, w was my position in assets 1
and 2.

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00:17:06,910 --> 00:17:11,130
So w1 was how many shares of asset 1 I
own.

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00:17:11,130 --> 00:17:14,150
And w2 was how many shares of asset 2 I
own.

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00:17:15,200 --> 00:17:18,390
And suppose I have another vector p1 and
p2

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00:17:18,390 --> 00:17:23,260
that are the prices of asset 1 and asset
2.

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00:17:23,260 --> 00:17:29,460
Then if I define v to be the dot product
of.

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00:17:29,460 --> 00:17:31,800
W and p so w dot p.

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00:17:33,040 --> 00:17:35,435
That's going to be the number of shares of
asset

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00:17:35,435 --> 00:17:39,710
1 times the price of asset 1 plus the
number

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00:17:39,710 --> 00:17:42,855
of shares in asset 2 times the price of
asset

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00:17:42,855 --> 00:17:45,610
2 and that'll just be the value of my
portfolio.