Hello.
Welcome to Week 5 for Mathematical Methods
for Quantitative Finance.
So the first four weeks, we did our review
of
Calculus and now we're going to shift
focus to Linear Algebra.
And this week's menu is going to be
Vectors.
A vector is just, you can think of it as a
point in a plane.
So it's just a pair of numbers x,y.
And then it also can be extended so you
could think of it as a point in three
dimensions, in which case it would be x,
y,
z that would be a vector of length three.
a point in the plane x, y would be a
vector of length two.
And so we'll start off just by coming up
or, or going over the basic addition
rules and a few other things for vectors.
Then in the second section, I'll go
through vector length.
So unfortunately, one of the things you
lose when you start to have points in
two dimensions is you can no longer say,
you know, x is less than y.
And so, we need another way to compare
magnitude so
we use the length of the vector to do
that.
And then I'll go and,
into just a little discussion about a
plane.
So a, a plane is basically just like a
line, but in three dimensions.
So it it ends up having length and width.
And then that's going to lead us into
Systems of Linear Equations.
And so in two dimensions, you can think of
a System of Linear Equations as just two
lines.
And what you're interested in finding is
the point where they intersect.
In three
dimensions, now we have to think of
planes.
So two planes would intersect in a line
and then a third plane would intersect
that
line in a point, and that will be
the solution to our system of linear
equations.
And then, we're going to start talking
about
how we can solve systems of linear
equations.
And that's going to bring us to bullet
point 4, Elimination.
And following from elimination, we're
going to end up being able to view the
problem we're trying to solve as a matrix
so a
matrix is just going to be a collection of
vectors.
And then we'll look at specifically
solving the problem Ax equals b.
So A is a matrix, x is a vector and x is
the unknown, and then b is another vector.
And I want to find the vector x that
satisfies this equation.
one way to do that is use an inverse
matrix, so that's all we'll talk about and
section 7.
And essentially if
A is just a number, x were a number, and b
were a
number than solving Ax equals b would
involve just dividing both sides by A.
And so inverse of a matrix, I think of
that
as just A-inverse times A times x will
equal x.
And then whatever's left on the right-hand
side would be the answer A-inverse b.
And it turns out that in solving this a
nicer way of
doing that involves factoring the matrix.
And this is going to lead us into next
week's lectures.
So there are a lot of a lot of useful
factorizations of
matrices that will make seemingly
difficult problems a lot easier to solve.
And then I'm going to finish up by talking
about something called the R Environment
for Statistical Computing.
And that's just a software program that
allows you to manipulate matrices and
vectors.
When we
start dealing with this many numbers, it's
no longer practical to
try and do the calculations by hand on a
piece of paper.
So it's really useful to have a little bit
of experience, at
least, with a computer algebra system that
can do that for you.