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[BOŞ_SES] Hello.

The previous section of a linear transformation

We saw that indicated by the matrix.

We have reached the team with this matrix equation.

The existence of solutions of equations,

one or infinite number of matrices is that if

We saw that with the order determined.

Now we want to see the basic operations that can be performed on these matrices.

They are given here to list here, respectively.

These are fairly simple tasks.

Here we saw earlier, a (x) = Y

We biliiy can be shown to be equivalent to the coefficient of transformation.

Now our work, that on conversions

showing the number of transactions to be translated into that matrix.

Conversions, the two conversion

A (x) B (x) is equal to the same, these two transformation is

We accept this definition and

subject them to all incoming equal to x,

opposing them in the matrix is defined in the same equation.

Now let's see how it will appear.

Now if you two conversion, A (x) B (x) are equal,

with their matrix representation A (x)

a ij xj B(x)'in de b ij xj olduğunu gördük.

Let us unite the two sides here.

Collection of the border are the same, XJ also common factor

aij bi minus multiplied by j XJ unite for equal

It will be zero and that this is true for all i,

We want it to be and for it to be valid values of all XJ

istiyosak it has only one solution, the coefficient is zero here.

Here, too, the coefficient of zero means that the matrix elements to zero.

So here we are removing the following results: Two transformations are equal,

these opposing matrices are equal.

I had an operation.

Or, as a matrix transpose Transpose

called a matrix.

This is shown by t from Transpose.

To transpose a ij we replace it with a jiu

when it comes to changing the column with the line of the matrix.

Ij is the ith element of this j'yinc of j'yinc

It comes to bring the i'yinc element of the line.

This pratice is very easy.

For example, let's start with a rectangular matrix.

1,2,3, -2,1,0.

We will do this line of the column.

We will make the first column of the first line, second line, second column will do.

It is available to the transpose (Transpose) call.

He has a theory.

And to transpose, t equals because that is the order of the inverse matrix

A number equal to the number of column t is independent of the individual rows.

We have seen, we have also proved but without the order of a matrix,

equal to the number of independent lines,

also equal to the number of independent columns.

So the number of columns in the independent line of the matrix equal.

E is inverted when you go to the column lines,

The independence of these columns will not change for you for making line,

We see that equal ranges.

Ie x conversion to multiply a matrix by a number

There are number of car you get hit by it with a new conversion.

C (x) say.

Could this C (x) 's opponents

depends on how each of the elements with the elements of that matrix?

We found it easily now.

To (x) opposing a matrix XJ j times.

We saw it.

We stood at c.

E c c means to multiply the means to multiply all this a ij of times.

Let us assume that c ij c ie opposing the transformation matrix.

c ij matrix.

C take it again indoors,

If we combine these two limits because the same collection,

Taking xj'y well as the common factor we find is equal to zero.

Only be possible with this equality does not apply to all XJ,

in the case where c is equal to c times a IJ ij'y

So when you get hit with a matrix of the elements of the matrix C,

Coming equivalent to a multiplication by car all the elements of the individual.

The sum of two matrices of these transformations again

converting matrix replacement.

(Xi) a ij the number of times when we show the XJ that this transformation,

BX Similarly, we do not know yet x is a c c c x j, j is X times.

In the same for all of them to collect the border

If we combine that with using common XJ,

we get the equation you see.

This can be provided for all x, but its coefficient is nil

possible, this new matrix, the oldest of the total matrix element

a and b are considered equal to the sum of the elements of the matrix DUğUylA come.

Now here comes out just a compliance requirement.

Matrix of rows and columns in order to collect the two should be the same.

Because the money is supposed to be a ij'y from a b j.

Three three-pointers a binary matrix with two Dola

You can not collect because the matrix corresponding to each of a ij'y is not a b j.

But three three-pointers in a matrix of a matrix can always collect three three-pointers.

This condition is also called compatibility.

How to multiply two matrices that mean?

This means that bump means applying successive conversions conversion.

b You receive the x, you can apply for a take on it.

Let's put into this matrix.

b x'e y desek b x eşittir

y'nin matrislerle,

ie where the number of such representation, always know, as you see.

b j XJ important in this order.

i where left and y index on the right side,

the index, the index also good.

There are also collection on j.

Let's apply this transformation to take on so au

So let's apply on au b x on y, z let's find a new.

The month is: flowing good.

I still here on the collection.

We're due to: Here we put here for that y y i.

This could be anything but short collection to be on i

because it occurred in our definition.

The second index, the indicator matrix of this y'yl,

With the indicators we take the same and we find zkan.

Now unite them.

How do we unite?

y'yi biliyoruz, y i'yi.

aka i'm writing here.

y is within the brackets with a gather,

This will be the sum of both the .alpha.

C, these letters are important here, the indicators.

on a collection jar.

Because here we've received j x, j are gathering on.

There's also a collection on the good.

im going at it, at the end of a short stay.

So here free index, the indicator has to be short.

There are also collection on j.

Again, we want to separate xj'y we want to know as we always do.

We can do this by changing the order of the collection.

See here are gathering on the jar before.

Then I got on fine collection.

If we change the order here, before we got on fine but collection

x of x i is not going out of this collection it.

She collected by merging the second, on how to collect the jar.

There is already gathering on the right side of the jar.

So we still would combine the two.

There are collection on j.

Ckj XJ common, but there's more of us coming here a second collection was.

Now there is only one remedy for it to be valid for all XJ.

He is zero term in the brackets.

This shows us that the aki bij Ckj'n.

The location of these important fevakala index.

See where the first index k, j last index.

On the right side the first index k already have one last index, j.

The difference index is repeated and collecting on it.

Therefore, a fine, the collection went on and lost an index.

The remaining indicators k and j.

Now here we have a compatibility condition.

K for that because I collect on

It showed the first display line.

The second indicator shows column.

b shows the reverse situation i row j free.

So, this collection to be compatible

the number of columns in a

It must be the same number of rows in b.

We wrote them visually.

Now we see that this collection of time

Considering where this k'yınc line.

Let's take for example three.

Here j'yinc this column.

Here it consists of a vector.

The first index, the index three,

wherein the second indicator consists of a column with five.

These two product actually comes from an inner product.

This vector is coming to the inner product of these vectors.

For this to happen, the number of milk in first second vector

It means the same as the number of lines in the vector.

The number of milk that genelletin of the matrix, the first matrix column

The number of rows in the second matrix is equal to the number of the collection will be available.

Otherwise it would be incompatible with this product and the individual

It shows that obtained by an internal multiply items.

So we do the process as follows: take a column of the second matrix,

Turn first matrix row with one individual stands.

We put them in their place.

We will see examples of this on the gene.

Coming equivalent to matrix multiplication.

One is X. multiplication and totals,

CAE must be acting on an x (x) you can find.

X b is the same on etkiletin of B (x) you can find.

A (x) and B (x) have collected by A (x) plus B (x) you can find.

We can also show the following: where x you give.

A plus B x you get.

That is where A (x) plus B (x) vs. came (A + B)

as if the X application processor built on.

If we try to explain it by shipping,

x impact on the A (x) we find.

At the same susceptibility x on B, B (x) we find.

We collect this when we both (x) plus B (x) is.

Instead, so let's call it a transformation that x

takes you directly to the (x) plus B (x), where you bring in, you bring it here.

We call on the sum of A plus B.

In situations like this product:

To (x) Do alıyos, giving a year.

This year, it will affect on the B z gives.

You diyosunuz Instead we get into such a processor and find that x, z it out.

This we call B.

Order is important, because here (x) you start with the coming year.

You etkiletiy B on it.

B is the second application.

Already here in the first factor, the second factor B.

There seems to be inverted as well as B somethin 'but operationally ago

Thinking here of (x) would begin.

(Xi) You are on B..

B the second process.

If you explain it in submissions,

You receive xi A'yl you convert, you'll find a year.

You apply to B on this y, z you find.

And you diyosunuz viewed Could I have such a new C

Can I find that you get X you take the processor directly here?

Here it also (BA) call.

[BOŞ_SES] Here,

This summary of what we have been talking.

bij with aija have collected by CiJ tactics.

bij'y with a hit of that,

j and k each of which already have collected by the

it is the product of a vector you secure.

This gives ckj.

This brings us to the product matrix.

He has a theory: A'yl to B.

Suppose we get it overturned hit.

This develops as follows: The product of this revolution, but in reverse order.

You shifting the column line because when you get overturned.

See, calculate it.

Let's look at the EU product.

There bij a collection of indexes on the middle and get lost.

k and j remains.

But that is also the inverse of the rational use of medicines.

at bji'nin devriğidir.

Here see them hit repeatedly, to stay away

in order that they may change for the index i, the number of them eventually.

Five UCLA, as well as the equivalent of three to five multiplied.

If we write in the supply of this type of BJ,

See you there on the middle collecting repeated index indicator.

Out there j and k.

Whereas in the beginning it is going to transpose it.

So this is equal to a transposable be transposed hit comes.

(EU) 's equivalent to taking it away comes transpose.

As though it's three, multiplied by 15 and you hit it, though matrix

After coming in the opposite equally collision took to get the inverse transpose individually.

This is an important theorem is often used in the calculation.

As a special vector, matrix type there.

We said we looked a vector matrix with a single column.

O a vector matrix with a single line.

But this relationship between the two lines of this column

Or, do as we've done just that your line column.

We call this column vector.

It is a column vector.

The row vector is going transpose it.

This vek, this column will be the line you take away the matrix transpose.

Or, in this line matrix

We call the vector line.

In this inner product of vectors in the plane and so on, we enter into these details.

We now have the product of x and y X. Or, we now have also multiplied by itself.

But in order to make it compatible with the need to use this matrix transpose.

For example, say you multiply x x transpose

It is a horizontal matrix of the first matrix,

line that matrix, we see that the second vertical matrix.

Matrix multiplication means, we take the turn hit the column here.

E which x2 x1 squared squared,

hence x length, giving the square of the length.

Çarpsak with y xt Similarly,

Or, horizontal row vector xT

matrix öbürkü column matrix.

Again, think of this product, we will turn to take to bring this column is horizontal.

We'll hit from opposite elements.

Burda hakkaten x (i) y (i) 's that it is equivalent

We see that the sum of the product of the year in which the x.

[BOŞ_SES] Now

I want to make examples.

Now, let us again sometime.

After crossing over them with just a little example of such proof

it becomes very concrete, we can easily see that it works is very simple.