# MOOC_LinearAlgebra_Lesson04

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Hello, and welcome to lesson
four of this Introduction to
Linear Algebra with Wolfram U.
The topic for this lesson
is Vectors in N Dimensions.
Let's begin with a brief
overview of the lesson.
what’s a vector?
You can think of a vector
as being an object, which
has both magnitude, like
a length and a direction.
Here are two vectors,
one of them is blue,
the other one is red.
And they each have a
length, which happens to
be different in this case.
And they have a direction
given by these arrows.
In linear algebra you think
of vectors differently.
You think of them as being
ordered list of numbers.
Each vector over here
is really going to be an
ordered list of numbers.
On the other hand, a scalar is
simply a number, it could be
positive, or negative, or zero.
Now, you can combine these
vectors by using a combination
of multiplication by scalar
in addition, and we get the
idea of a linear combination.
given another vector
does it have the property
that it can be expressed
as a linear combination
of the given vectors?
That leads to the notion
of vector equations
and linear systems.
So that's the plan for this
lesson talk about vectors
and vector equations.
Let's begin with a simple case
of vectors in two dimensions.
Vector in two dimensions is
just a list of two numbers.
For example, you got
a vector, 3 and -1.
It's got two entries over here.
Let’s define it, like
MatrixForm to it, you
get back a column vector.
You think of these
vectors as being really
columns of a matrix.
They’re column vectors and if
you put them all together, then
the set of all vectors in two
dimensions has got a special
notation, it’s called R^2.
That’s written as double
struck capital R, it’s a
special symbol, a mathematical
symbol, which is worth knowing.
when are such vectors equal?
They’re equal if the
corresponding entries are equal.
For example, the two vectors
over here are not equal,
because the first 1 is 4,
7, the other one is 7, 4.
Let’s check that over here.
If you ask the question,
is v=w, using the double
equal sign, you get back
a false, which says that
those vectors are not equal.
Now let’s try and combine the
vectors in two dimensions.
There are two operations
the scalar multiplication,
namely multiply every entry
by a scalar, and there’s
addition, simply add the
corresponding increase.
For example, you’ve got two
vectors u and v over here.
u is 1-3, v is 2-7.
The question is: find 3u, find
-5v, find u+v and find 2u-3v.
Let’s first define them and
then what you can do is, you can
work out 3*u, just multiply each
entry by 3, so you get 3, -9.
You can do -5v, just multiply
each entry by -5 for v.
You can add the
entries to get u+v.
The last one is a bit more
difficult because you really
should do 2u, then do 3v,
then subtract them, but in
fact, it's quite easy to
do in the Wolfram Language.
So that's how we combine
vectors using scalar,
multiplication, and addition.
Now of course, because we're
in two dimensions, one can
talk about the geometry of
vectors in two dimensions.
What you can do first of
all, is set up a rectangular
coordinate system in the plane,
and then a vector is simply
going to be the same as a point.
For example, over here you
have a set of points on the
left and a set of vectors
on the right, and each
point over here corresponds
to a vector over there.
If you’re given two vectors,
u and v, then u+v really
corresponds to the fourth
vertex of a parallelogram.
In this picture over here,
you’ve got a parallelogram
with four vertices.
There's the origin, there's u,
there's v and then there’s u+v.
Those are the vertices
of the parallelogram.
Having defined vectors in
two dimensions, let’s get a
little ambitious and talk about
vectors in three dimensions.
A vector in three dimensions
is just a column vector, column
matrix with three entries.
Like before, they are defined
geometrically by points,
with arrows from the origin.
For example, here's a
vector in three dimensions,
2, 3, 5, that’s a.
Then 2a is just a vector
along the same direction
but twice as long.
Having talked about two and
three dimensions, we can
now go ahead and talk about
vectors in n dimensions, where
n is any positive integer.
Could be 2, could be 3, could
be 10, whatever, and that's
the space R^n, a very important
mathematical construct.
So R^n denotes a set of all
vectors in n dimensions.
You can think of each vector
over here as being a collection
of scalars u_1, u_2, to u_n.
It's just a column
vector with n entries.
The special vector called
a zero vector, which is
just denoted by zero.
Again, you can combine vectors
using scalar multiplication
or using addition.
The set of all these
vectors with these
operations is referred to
as the vector space R^n.
Vector spaces are just
collections of all vectors
for a certain type, with
the operations defined by
scalar multiplication and
addition, and the most
important vector space for
this course is the space R^n.
A simple example, here
is a pair of vectors in
four dimensions, u and v.
You can combine them, you can
do u+v and you can do 3u-7v,
and it's all quite simple.
That's the idea of a vector
space and now we can talk
about linear combinations.
Suppose you’re given a fixed set
of vectors, v_1, v_2, up to v_n,
n could be 3 or 4, or whatever,
and scalar c_1, c_2, up to c_n.
Then what you can do is,
you can combine them.
You can first multiply each
vector by the corresponding
scalar and then you can
add them up and you get
a linear combination of
those vectors, with the
weights c_1, c_2, up to c_n.
For example, here are the
same vectors u and v from the
earlier slide, and you can
let’s say do √3*u+v, which
just says multiply each entry
of u by square root of three,
then add it to v, and you get
back the result over here.
Or you could do ½*u and
0*v, that just gives
you back one half of u.
Or could do zero times each
vector, which of course, gives
you just zero as a result.
That's about fixed vectors
and fixed scalars but what
would happen if you actually
took a fixed set of vectors
over here and then try to find
all the linear combinations.
Well that's called the
span of those vectors.
Basically you take the
collection of all vectors,
which are linear combinations of
this particular set of vectors.
Now that looks really
abstract but for example,
if you have a single vector
v, then that’ll give you
a line through the origin.
For example, use a vector v
and you multiple it by c’s and
you get back a straight line.
Then on the other hand, if
we’ve got a pair of vectors,
u and v, then you get back a
plane because you couldn't do
all the multiples of u over
here, all the multiples of v
over here, and then you can add
them up and get back a plane.
So that's the notion
of a span or subset.
suppose you’re given an
arbitrary vector, which is
not one of those, suppose
you're going to vector b, can
you somehow combine them as a
linear combination of the views?
Here's your v_1, v_2,
given to you and it's a b.
can you actually check
whether b belongs to the
span of those vectors or not?
A span in this case
is just a plane.
You want to figure out whether
b is in that plan or not.
The question is: does this, over
here, have a solution or not.
To do that, what you do
is set up the augmented
matrix for the system.
You've got 1phi, -3, 2, -13, 8,
and 3, 3, 2, because you got 1,
5, -3 over here, 2, -13, 8 over
there, and 3, 3, 2 over there.
What you do is you
do a row reduce.
Then you see over here that the
last row is 0=1, that’s no good.
That says that this system over
here has got no solution, and
hence, the given vector b is
not in the span of v_1 or v_2.
Somehow starting off with
these spans will come to
linear systems and vector
equations and that's how we
determine if a particular
vector b belongs to span or not.
That brings me to
the end of lesson.
The point is in this
lesson, you've learned how
to represent vectors in n
dimensions, not just 2 or
3, but any dimensions, and
they are just column vectors.
We also talked
about vector spaces.
The collection of all
vectors in R^n and linear
combination of vectors.
And then given a subset
of vectors, you can
talk about the span.
Then the question is:
how does that relate
to linear combination?
That's just a collection of
all linear combinations and
that leads to the question of a
vector equation, every time you
want to check whether a vector
b belongs to the span or not.
That's the end of this
lesson and in the next
lesson we’ll talk on matrix
equations, but before we
do that, do this important
lesson, it's quite abstract.
Do go over it carefully and
be ready for matrix equations.
I’ll stop over here.
Thank you very much.