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[BOŞ_SES] a

Our previous session,

We examined the issue together linearly dependent and independent vectors.

Anti vectors i and j in this plane and its side

for example a vector derived from the sum of i and j may also be considered.

They would be dependent.

Now we start with the following concepts: Linear expansion.

Contact a e1, e2, e3, get confident team given vector.

These vectors c1, c2, c3,

You can get hit with gathering a vector x cm coefficients.

Obviously, the coefficients for variables c1 to cm.

We take different as we dealt with each change a new x

We'll get vector.

Therefore, by changing this set of vectors x c it is formed.

Here in this cluster formed by the vector x

We call linear expansion of the relative position with respect to this email.

This is the way to hit the vector x by collecting a set

and c are obtained from our newly formed you change vectors

expansion of the set of vector e, we call linear expansion.

Just following simple theorem is obtained logic behind that.

e1, e2, linear expansion of the team soak create a linear vector space.

Where x is a set of very clear that now.

Unless you change team every car we get a new vector.

The whole point of this vector x, so that x is the vector of gathering and

be open or closed under multiplication by a number of clusters.

Other words a more obvious if we take a consisting of a x c y we take it that d1,

d2, d3, whether DM, when we get these two balls

new z vector of the vector sum of the two also he finally e1,

e2, e3, is obtained from EM opening in this way.

This set is closed under vector sum of these kind of means.

More clear words, when we take two vectors

obtained after summing these vectors are within the same cluster.

That can be expressed in this email are.

Again, this vector x çarpsak with an alpha number,

Once the vector obtained alpha alpha times c1 c2

vector e is obtained by combining the alpha coefficient times cm.

So in each of these vector multiplication by a number

It remains under within the same cluster.

The cluster will be closed.

If you remember the theorem we have seen, under these two processes

which it closed sets form a linear space.

Of course, we see that linear space a superior structure of a cluster.

Because no transactions in the cluster.

There are only elements of the cluster, there are relationships between them.

Give an example of it right away.

The plane e1, e2 and e3 given vectors we take,

so hit them with their linear expansion coefficient

we obtain a set of vectors are vectors in the plane again.

One e4, e5, e6 again as we take the vector,

of vectors in the plane, their linear expansion is plane.

A linear space.

But this linear space are plane.

On the other hand we take any two of these,

e1 and e2 not all three of them if it works i and j,

linear expansion thereof is plane.

Therefore, the following question comes to mind: it's the economy,

How can we be the most economic one open space.

Then we call the base team vector tool makes this process.

If we consider the example I just plane and a plane j vectors

base creates.

Similarly, the E1 and E3 constituting a base.

e2 and e3 constitutes the base.

One e4, e5, e6 you take in twos they can create base.

I say you can create.

Because you take this e1'l e1 instead of three zero three zero vector does not create this base.

Because it is interdependent.

Now we call the smallest team in the bottom team in these vectors.

Or called in Turkish bases.

A, at a given space that can be opened

The base of the vector space is called a small team.

Just these two theorems follow one another.

It is linearly independent vectors of a bottom pack.

Because even if one of the others expressed linearly dependent

now you could have a smaller team because you can.

Therefore, if they were not the smallest initial dependent.

Therefore, regardless easily it is revealed immediately following the definition.

And yet it all in the following theorem vector base

The number of independent vectors of the base are the same.

This again comes from the definition of the base and team

the number of vectors is called the size of this space.

These definitions and theorems follow one another.

If they were so interdependent base team claims

Watch your team in the vector,

you expressed when one of öbürküsü dependent.

Smaller he goes.

Therefore, it would be less than the number.

Now, let's make an example again.

We saw in the previous example.

e1, e2 and e3 plane vectors are linearly dependent.

We take any two of them in their time

It is independent and therefore can form a base for the team plane.

All at any level in any two of them and two of them

It is sufficient to vectors.

The same vector twos to work for it

representation by vectors such new coefficients Y1 base,

Y1 Y2 base or two bases, including two base Y2 shown.

But here's the important number of vectors base team

the number of all vectors is two.

This also shows the size of the space.

Shows that two of the plane's size.

Let's take an example of a little more complicated.

We have seen examples of this.

We see here that it is linearly dependent.

Rather, we see examples like this.

When we look at them, c1, c2, c3 three vector write

c3 by the number zero in eşitleyin here,

We find the coefficient is zero.

If c2, c3 zero is zero.

c2, c3 If zero is zero.

c1 is zero.

If you look here, this is zero.

So the only solution of three equations for three unknown c1, c2,

c3 can only see the equation can be effected by not zero.

They are therefore independent.

Therefore, e1, e2,

e3 vectors form a base.

However, since their number is three, this r4

Although the three-dimensional space has reached the space, it produces.

So it generates a subspace of R4.

Here, too, there is an important observation as follows.

Often it is a mistake like this.

You count the number of components here.

You call this four-dimensional space.

However, defining the size e1, e2, e3 basis vectors

It is that many of their number is not defined by the number of how many.

For example, a number of them, although they are defined by the number of four three

because they produce three-dimensional space.

This is an important observation but a simple observation

It can also issue to be considered in the same function space.

Of course, this is not a function such as a vector to a cosine square e1, though,

Although sinus square e2, e3, though the cosine of 2x

2x right here from cosine cosine squared x

plus or minus sine squared minus the E1 and E3 that we have to know

we see can be expressed in terms of e2.

This means that three linear expansion of a function space

Although the three constitute a base.

Because one of them dependent on others.

If you take any two of these vectors in which a two-dimensional

function space, space that creates a special function

base of infinite infinite-dimensional function spaces

It produces a two-dimensional subspace.

Again, let's take an x and x squared function,

they are separate from each other because all possible

x values for the c1 times a plus

times c2 and c3 x times x squared 0

it should be a given c1, c2, c3 You may be able to.

In the end, all right, but a second degree equation c values

To solve Cantabile possible to solve for x c values do not

and it is also the only solution to 0 c1, c2 0, c3 is zero,

Therefore, this function is a three-dimensional

producing a space.

Second degree in this space

or less, e.g., one of the following

If the primary function can produce zero zero.

Function of a three-dimensional space can be produced from a combination of these coefficients.

Again, a three-dimensional subspace of an infinite-dimensional function space

We see that.

Go to the infinite dimensions of the situation somewhat more

confused because I explain it with an example.

Let's give the JX j cosine function data we had zero zero one,

j equals 1, j equals two, plus, minus, can put forever

Let them go so that linear spaces,

When we take the linear combination can produce an infinite dimensional space

but can not produce it because all functions and its cosine x

the floor is a double function for x in multiples functions,

x minus x to be just the same for exchanging a couple of functions that

whereas the general functions of symmetric functions can produce them

There are infinitely many functions due to the non-symmetrical function in space

If we find that the base vector has found independent of each other vector

Or, in this case their function if we find that if we base

You can create as anti unsymmetrical

ALSA symmetric functions that you will not denominated in cosine

But if it manifests itself in problems for you

just because we want to produce a truly symmetric functions

also set us all a dual function of the cosine function jx

data excludes only functions but also ruled out because of our function

What if neither symmetrical nor anti-symmetrical double excludes this does not create a database.

All vectors in order to establish base in all functions

You must be able to show.

Now that immediately follows the concept vectors

whereby components of a linearly independent vectors

They are working with the team because they form the base of any

We express Exorcist them from a vector x vector.

Given that this issue immediately following the team base

by the x1, x2, xn coefficients is one.

It's easy to prove this, the vector x x1

x2, xn GB coefficient and also the base x1, x2 base,

Show the factor we can assume that the same base as x3

x vector to show two different component, I can assume,

As you can see them from each other çıkarınca çıkarınca left side to give zero,

where x1 x1 minus minus x2 x2 base as base

As we get this equation xn xn minus base.

E1 E2, e3, that is independent at each other linear

The only solution in the composition is zero coefficient e1, e2 of the coefficients is zero,

E3 coefficient is zero coefficient of EN.

EUR X, which indicates that the coefficient is equal to x üssül coefficient.

So our initial assumption is not correct.

x1 x1 and base have to be the same.

x2 x2 base and therefore have to be the same x in only one

There representation.

Since this is a single number that we give a name BUNLARADA,

We call it anything, the component must be considered as components

According to the team, but only the base under the floor of one team.

Components of the base team will change you replace this component

Let us explain this with an example as something quite natural.

Let the vector X one and three.

No matter their rectors plane E1 and E2,

independently therebetween and two-dimensional vectors

one and three components of this vector can show them all.

Because we write in terms of x when they're getting it.

This plane

we show the vector in the plane, as you can see here a third vector.

This two component toward a j direction component of the three.

But the team that we change the base e1, e2 instead of f1

and f2 if we are now easy to see that they are independent from each other.

They form a base plane because two

a base vector, they may be in line with each other as long.

Independently from each other when the two vectors form a base.

According to this base see show xi team

the same vector in this new vector tool here

We see that this time was different components.

This b1 and b2 are previously've found that the components of one and three,

where the components of the two, and we find it.

This is a tricky thing, something amazing,

According to one but not anything surprising because the base components of the team,

According to the team to have a single, one and three; According to one team still had fun,

components of furniture, as we have an account here, and they are two.

This account is very easy as you can see we f1 and f2 of x

We want to write it as a component.

given an f1, f2 given a negative one.

To them, when we held the first component b2, b1 will put a minus.

B1, B2, plus three to give.

Two unknown two equations.

It is going b2, as you can see, when we collect, four would be twice b1,

two would b1, b2 b1 after finding the

here as a minus

obtained.

[BOŞ_SES] [BOŞ_SES]

Let's do an example for similar functions.

FX function of x squared plus you get three,

this second moment in the space of quadratic functions

and a function in the lower-order functions.

We have this base here, we take a xx a base frame,

components in the base of these three functions,

For that there is a zero and a square where x is x.

But we can also select a negative xx x squared plus team well.

You can easily show that they are independent from each other.

This base set of three terms plus

here it is a time of x squared plus x b1,

b2, b3 times one minus x times x squared plus he wrote

If we find these three parts of the two coefficients b,

We see that two and three split out, in fact we can do to provide here.

That means the same function in this new base team

Although the components are different from the first one.

[BOŞ_SES] An example is given here again.

We can proceed.

Now again we would like to take a break here.

So we reached the base concept using the concept of linear dependence.

We reached the base concept of using components concept.

So far we have seen a number of the collection and processing of shock

We also know from the plane, the angle between the angle between two vectors

We had to find a new processing requirements, the inner product that cosine

Open the middle gave, even to this public space forever

genelletil the dimensional function spaces in the inner cross to still be with.