# MOOC_LinearAlgebra_Lesson04

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Hello, and welcome to Lesson 4
of this Introduction to Linear Algebra
with Wolfram U.
The topic for this lesson is
vectors in <i>n</i> dimensions.
Let's begin with
a brief overview of the lesson.
The question is:
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.
But in linear algebra,
you think of vectors
slightly 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 scalars
and addition,
and we get the idea of
a linear combination.
The final question is:
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.
OK, so, let's begin with
a simple case of vectors in 2 dimensions.
A vector in 2 dimensions
is just a list of two numbers.
For example, here you've got
a vector, 3 and -1.
It's got two entries over here.
Let's define it.
You apply 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 2 dimensions
has got a special notation;
it's called <b>R</b>^2.
That's written as
a double-struck capital R;
it's a special symbol,
a mathematical symbol,
which is worth knowing.
The question is:
when are such vectors equal?
Well, they're equal
if the corresponding entries
are equal.
So for example,
the two vectors over here
are not equal,
because the first one is {4, 7},
the other one is {7, 4}.
Let’s check that over here.
If you ask the question,
is <i>v</i> equal to <i>w</i>,
using the double equals sign,
you get back a False,
which says that
those vectors are not equal.
Now, let's try
and combine the vectors
in 2 dimensions.
There are two operations:
the scalar multiplication,
namely, multiply every entry
by a scalar,
and there's addition,
simply add
the corresponding entries.
For example, you've got
two vectors <i>u</i> and <i>v</i> over here;
<i>u</i> is {1, -3}, <i>v</i> is {2, -7}.
The question is: find 3<i>u</i>,
find -5<i>v</i>, find <i>u</i> + <i>v</i>
and find 2<i>u</i> - 3<i>v</i>.
Let's first define them,
and then what you can do is
you can work out 3<i>u</i>;
just multiply each entry by 3
so you get {3, -9}.
You can do -5<i>v</i>,
just multiply each entry by -5
for <i>v</i>.
You can add the entries to get
<i>u</i> + <i>v</i>.
The last one is a bit more difficult
because you really should do 2<i>u</i>,
then do 3<i>v</i>, then subtract them,
but in fact, it's quite easy to do
using the Wolfram Language.
So that's how we combine vectors
using scalar multiplication
and addition.
Now, of course, because we're
in 2 dimensions,
one can talk about
the geometry of vectors
in 2 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,
<i>u</i> and <i>v</i>,
then <i>u</i> + <i>v</i>
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 <i>u</i>, there's <i>v</i>
and then there's <i>u</i> + <i>v</i>.
Those are the vertices
of the parallelogram.
Having defined vectors
in 2 dimensions,
let's get a little ambitious
and talk about
vectors in 3 dimensions.
A vector in 3 dimensions
is just a column vector,
a 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 3 dimensions,
{2, 3, 5}, that's <i>a</i>.
Then 2<i>a</i> is just
a vector along the same direction
but twice as long.
Having talked about
2 and 3 dimensions,
we can now go ahead and talk about
vectors in <i>n</i> dimensions,
where <i>n</i> is
any positive integer,
could be 2, could be 3,
could be 10, whatever,
and that's the space <b>R</b>^<i>n</i>,
a very important
mathematical construct.
So <b>R</b>^<i>n</i> denotes
a set of all vectors
in <i>n</i> dimensions.
You can think of
each vector over here
as being a collection of scalars
<i>u</i>_1, <i>u</i>_2, up to <i>u</i>_ <i>n</i>.
It's just a column vector
with <i>n</i> entries.
There's a special vector
called a zero vector,
which is just denoted by 0.
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 <b>R</b>^<i>n</i>.
Vector spaces are just
collections of all vectors
of a certain type,
with the operations defined by
scalar multiplication and addition,
and the most important vector space
for this course
is the space <b>R</b>^<i>n</i>.
A simple example:
here is a pair of vectors
in 4 dimensions, <i>u</i> and <i>v</i>.
You can combine them—
you can do <i>u</i> + <i>v</i>
and you can do 3<i>u</i> - 7<i>v</i>
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,
<i>v</i>_1, <i>v</i>_2, up to <i>v</i>_<i>m</i>,
<i>m</i> could be 3 or 4 or whatever,
and scalars <i>c</i>_1, <i>c</i>_2,
up to <i>c</i>_ <i>m</i>.
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 <i>c</i>_1, <i>c</i>_2,
up to <i>c</i>_ <i>m</i>.
For example, here are
the same vectors <i>u</i> and <i>v</i>
from the earlier slide,
and you can, let's say,
do √3 • <i>u</i> + <i>v</i>,
which just says,
multiply each entry of <i>u</i> by √3
then add it to <i>v</i>
and you get back
the result over here.
Or you could do
½ • <i>u</i> and 0 • <i>v</i>;
that just gives you back ½ <i>u</i>.
Or you could do
0 times each vector,
which, of course,
gives you just 0 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 tried 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 <i>v</i>,
then that'll give you
a line through the origin.
For example, here’s a vector <i>v</i>,
and you multiply it by <i>c</i>'s
and you get back a straight line.
And on the other hand,
if we've got
a pair of vectors, <i>u</i> and <i>v</i>,
then you get back a plane
because you can do
all the multiples of <i>u</i> over here,
all the multiples of <i>v</i>
over there,
and then you can add them up
and you get back a plane.
So that's the notion
of a span or subset,
and the question now is:
suppose you're given
an arbitrary vector,
which is not one of those.
And the question is:
suppose you're going to vector <i>b</i>,
can you somehow combine them
as a linear combination
of the <i>v</i>'s?
For example, here's your <i>v</i>_1, <i>v</i>_2
given to you, and that's a <i>b</i>.
The question is:
can you actually check
whether <i>b</i> belongs to
the span of those vectors or not?
A span in this case
is just a plane.
You want to figure out
whether <i>b</i> is in that plane or not.
The question is:
does this have a solution or not?
To do that, what you do is
you set up the augmented matrix
for the system.
You've got {1, 5, -3},
{2, -13, 8}, and {3, 3, 2},
because you've 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 <i>b</i>
is not in the span
of <i>v</i>_1 or <i>v</i>_2.
Somehow, starting off with
these spans,
we have come to linear systems
and vector equations,
and that's how we determine
if a particular vector <i>b</i>
belongs to a span or not.
That brings me
to the end of the lesson.
The point is, in this lesson,
you've learned how
to represent vectors
in <i>n</i> 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 <b>R</b>^<i>n</i>
and linear combinations 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 combinations?
Well, that's just a collection
of all linear combinations
and then that leads to
the question of a vector equation,
every time you want to check
whether a vector <i>b</i>
belongs to the span or not.
That's the end of this lesson,
and in the next lesson
we'll talk about matrix equations,
but before we do that,
do 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.