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Hello.

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Welcome to Week 5 for Mathematical Methods
for Quantitative Finance.

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So the first four weeks, we did our review
of

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Calculus and now we're going to shift
focus to Linear Algebra.

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And this week's menu is going to be

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Vectors.

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A vector is just, you can think of it as a
point in a plane.

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So it's just a pair of numbers x,y.

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And then it also can be extended so you
could think of it as a point in three

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dimensions, in which case it would be x,
y,

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z that would be a vector of length three.

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a point in the plane x, y would be a
vector of length two.

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And so we'll start off just by coming up
or, or going over the basic addition

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rules and a few other things for vectors.

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Then in the second section, I'll go
through vector length.

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So unfortunately, one of the things you
lose when you start to have points in

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two dimensions is you can no longer say,
you know, x is less than y.

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And so, we need another way to compare
magnitude so

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we use the length of the vector to do
that.

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And then I'll go and,

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into just a little discussion about a
plane.

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So a, a plane is basically just like a
line, but in three dimensions.

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So it it ends up having length and width.

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And then that's going to lead us into
Systems of Linear Equations.

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And so in two dimensions, you can think of

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a System of Linear Equations as just two
lines.

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And what you're interested in finding is
the point where they intersect.

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In three

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dimensions, now we have to think of
planes.

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So two planes would intersect in a line

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and then a third plane would intersect
that

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line in a point, and that will be

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the solution to our system of linear
equations.

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And then, we're going to start talking
about

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how we can solve systems of linear
equations.

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And that's going to bring us to bullet
point 4, Elimination.

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And following from elimination, we're
going to end up being able to view the

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problem we're trying to solve as a matrix
so a

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matrix is just going to be a collection of
vectors.

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And then we'll look at specifically
solving the problem Ax equals b.

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So A is a matrix, x is a vector and x is
the unknown, and then b is another vector.

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And I want to find the vector x that
satisfies this equation.

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one way to do that is use an inverse

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matrix, so that's all we'll talk about and
section 7.

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And essentially if

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A is just a number, x were a number, and b
were a

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number than solving Ax equals b would
involve just dividing both sides by A.

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And so inverse of a matrix, I think of
that

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as just A-inverse times A times x will
equal x.

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And then whatever's left on the right-hand
side would be the answer A-inverse b.

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And it turns out that in solving this a
nicer way of

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doing that involves factoring the matrix.

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And this is going to lead us into next
week's lectures.

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So there are a lot of a lot of useful
factorizations of

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matrices that will make seemingly
difficult problems a lot easier to solve.

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And then I'm going to finish up by talking

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about something called the R Environment
for Statistical Computing.

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And that's just a software program that

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allows you to manipulate matrices and
vectors.

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When we

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start dealing with this many numbers, it's
no longer practical to

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try and do the calculations by hand on a
piece of paper.

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So it's really useful to have a little bit
of experience, at

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least, with a computer algebra system that
can do that for you.