Understanding Vectors - Practical Machine Learning Tutorial with Python

what is going on everybody and welcome
to the 21st machine learning with Python
tutorial video we're have been talking
about the support vector machine in the
previous tutorial we talked about the
high high level intuition of a support
vector machine and then we actually
applied it with real data and now what
we're going to start doing is breaking
down the support vector machine but
first I want to talk about vectors of
pretty much integral hence the name
support vector machine so I want to talk
about a few things about vectors just to
make sure everyone is on the same page
so let's go ahead and get started so
let's draw a vector so first let's have
some we're going to have some vector
space here otherwise you can call this a
feature space as well so this dimension
here coming out might be your x1 and
then this dimension here might be your
x2 so you might call this Y or your
feature space or your vector space now
let's go ahead and add some ticks here
so 1 2 3 4 5 and then 1 2 3 4 5 and now
let's actually draw a vector so first
let's actually what will define the
vector first so we're going to say
vector a and the way you denote a vector
is with the letter and then above it
you'll have an arrow some people would
just put a bar and then a lot of times
people forget to put anything at all so
if it's just a bar sometimes you're just
supposed to use your intuition to figure
out it's a vector and then sometimes
they just forget to put it I might
forget to put it just understand when
we're talking about vectors well we're
talking about vectors so anyway vector a
and whereas a vector a is equal to 3/4
and a vector has both magnitude and
directions so let's go ahead and plot
out vector a so to do that it's just
like a Cartesian coordinate at first
anyway so 3/4 so about right there and
that would be our vector a that's our
basically our direction because we know
vector a is 3/4 from origin so origin is
just 0 0 and so I'm going to get rid of
that but I just wanted to circle it so
origin 0 0 and we will draw an arrow out
so that is
Ector out two three four so we know the
direction now how about the magnitude so
the magnitude is denoted like so double
bars around the vector you'll see a lot
of people just use single bars I don't
think anybody's going to use no bars
that would be too confusing but anyways
single bars double bars means the
magnitude and the magnitude is the same
as the norm is the same as the length
we've actually already calculated the
norm if you recall with Euclidean
distance with our K nearest neighbors
it's the exact same algorithm that we
would use so just keep that in mind but
let's go ahead and calculate in this
case how might you calculate the
magnitude of vector a how do we do it
well to calculate the magnitude of a
vector it's just the constituents of
that vector squared I guess actually the
way I would describe is the square root
of the squared constituents summed
together and so basically let's say we
wanted to figure out the magnitude of a
that would be the square root of 3
squared plus 4 squared that would give
us 9 plus 16 otherwise known as 25 the
square root of 25 equals 5 therefore the
magnitude of a equals 5 so that is the
magnitude and if that looks familiar
that's because it should be familiar
because of life I grew in serum we could
make a triangle out of this right so you
could say this is a side of the triangle
this is a side of the triangle this
length is 3 this length is 4 3 squared
plus 4 squared I'm still 25 and if you
recall Pythagorean theorem a squared
plus B squared equals C squared that's
basically a restatement of this right
here so yeah it's the same thing the
only difference is we can continue
adding dimensions out to calculate the
magnitude so now the next thing I want
to talk about is the dot product so with
that product this is the dot product
between two vectors so let's say you've
got a vector a and the vector a is equal
to we'll do 1 comma 3 and then we're
going to say we've got a vector B which
is going to be equal to a stay for two
so to do the dot product you would write
that out is it's literally just a dotted
with B and generally here you'll see the
vectors and the way that you do it is
it's just the basically the constituents
times the same placed constituent in the
other vector so it would be 1 times 4
plus 3 times 2 so a dotted with B would
be 1 times 4 plus 3 times 2 so that
would be 4 plus 6 otherwise known as 10
so a dot B would be equal to 10 and of
course this is a scalar value so those
are the two major things that we
absolutely have to cover as far as
vectors are concerned because both of
these are going to play an absolutely
integral role in the support vector
machine as we start to build it up so
anyway if you have any questions or
comments or whatever up to this point
leave them below otherwise in the next
video we'll actually start breaking down
the support vector machine using vectors
and dot products and all kinds of fun
stuff so anyways stay tuned the next
video thanks for watching