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[BOŞ_SES] Hello.

Linear Algebra I and II of the key issues in the course of our matrices.

Linear Algebra I told the matrix going to the basic structure.

Linear spaces and linear processors

but here we do some simple calculations associated with matrices.

In the second part, Linear Algebra II

We see operations related to the matrix completely.

Equations, determinants,

eigenvalues of matrix inverse matrix, eigenvectors,

We see fundamental issues such as diagonalization of the matrix.

Of course the substance of these issues,

not able to do calculations without understanding the basics.

Furthermore, as MATLAB is a software,

today the most active, the most widely used software.

This supremely effective software to make calculations about the matrix.

Therefore, there is also interest in learning it.

Therefore, what you need to define the basic structure to be solved,

MATLAB also need to be resolved, but how it will complement our strength in our muscle

so we can have an important tool to make this work.

MATLAB word, as you can see here

'MATRIX LABORATORY' made of shortened words.

The main purpose of a software matrix.

Already a computer can not constantly definition

It works with discrete values.

Whether for its continuous function of these integrals, derivatives,

basic problems such as differential equations

Unraveling already rotated matrix.

He respects the MATLAB software that can work in all areas.

This is happening in many programs as an input, an output is happening.

But this is just an interactive software.

That is an interactive software.

You Yazıyos answer comes immediately.

Maybe too big a problem though, several times to open and close your eyes

but without the simple may need time to open and close your eyes here,

Coming answers to short problems.

The main structure of you here, we see the theoretical structures

problems and Linear Algebra I and Linear Algebra II

of course we solve analytically the main text

We will solve problems using MATLAB software.

A reminder: Turkish original three for MATLAB written in English,

S, G, O, U, they, and letters can not be used as their capital letters.

But this does not remove the difficulty.

You'll see.

The first step description of how it is done.

Matrices, vectors, matrices, or vectors row

column matrix structure can be and we want to start with matrix operations.

First thing,

how it is written in a line matrix.

First, the writing gap when you accept this as a new number.

Number 5, number -3, 10 is a number.

Gaps are important.

This also shows that it is a comma overhead line.

When you wrote that it appeared as follows.

As a line of matrix is the gap.

The first element of the row, the second element,

Just the opposite of that of course there is a third element of the matrix line.

Column matrix.

The basis of a matrix column matrix.

The first line, so you spend a semicolon,

When you see the line put into writing space here.

When you put a semicolon in column writing.

Five of writing as the first item in the column.

After the number again and point gap separated by commas

as the second element in the column writes and writes on the third item.

These rows and columns.

Same thing will happen in the matrix.

See, here the line as well,

As well as the existing line matrix column matrix composition.

Gaps are taking elements of the line.

1, 2, -3 matrix of the first, second, is the third item.

You separate the semicolon second line,

Disconnect the semicolon see you again as a third line.

Writing is one of the issues that we need the most basic learning matrix.

Now that dotted the hill just use another comma

transpose the matrix is the transpose.

That knocks the column matrix, line matrix doing.

Line matrix knock it, making the column matrix.

That line column, turning column line.

Here on the hill

Or, comma top,

mark indicates that this is the base line.

When you remove the column that was it.

Now that r hill

If you select a comma, this line is the matrix transpose.

That makes this line matrix column matrix.

Numbers written the same but the column matrix.

So we learned that one of the processes equal and semicolon

Or, the base of this hill outside the comma sign.

Similarly, if you receive the üssülü h, s column matrix we see here.

This üssülü, its transpose it knocks.

Line writes the matrix.

Equal is, taking the view that defines the r right.

It describes itself as s he wrote right here.

He describes it as a summer right here.

Or, as a third of a second agent also define b.

See gene (1, 2, 3, 4, 5, 6, 7, 8, 9).

This is the first line, second line, third line.

Semicolon separates the rows.

Gaps in the line separating the elements.

See exactly where B is 1, 2, 3, 4, 5, 6, 7, 8, 9 written.

However, when we put the base account is the matrix transpose.

Lines column, replacing the columns line.

The same thing as B, B base as TBA

T here that he remembered so you transposed he transposed T're remembered.

You are completely free to make definitions on the left.

However, this software has its own unique icon.

No matter where you put brackets instead of parentheses normal.

He would not accept.

Anything else understands.

If you want to write matrix brackets,

because it is something that will remain easy in mind.

B is üssülü account, we write it as TB.

See here.

Now that may come to mind immediately.

A symmetric matrix, we know that the Transpose,

equal to the matrix transpose.

So we say we remove from C tau,

if the symmetric matrix C becomes zero matrix.

Because and TE equal to each other.

If not, it is not zero.

To check these two,

And that B is symmetric, D to define a well.

B equals minus TB in D say.

You can see see that C has zero.

This intermediate process we do not see anyway.

It gives us is removing the matrix software matrix.

Looking to D is not zero.

Just understand that here,

The difference was because the symmetric matrix transpose zero.

B matrix is not symmetric.

The first they learned processes.

The second kind of process related to the matrix, the matrix to be multiplied by a number,

matrix, summing the two matrices, multiplying a certain ranks of two matrices.

We know that A plus B matrix, the matrix A is equal to B-plus.

But the matrix A times B is not equal to the BA strikes.

Here I want to draw more attention to something new.

we give a title that also dispersed.

This made the process of what we might want to remember that.

Then we write such a title.

Matrix operations, Example 1.

And of course it is not evil or even dotted the i in English

Examples of these writes, transaction writes.

But it also does not have a difficulty in understanding one.

as well as the dispersant "Display" comes from the word.

The display them, says you type them.

When you give this as input, as it turned out, as the title says it already.

You remember that time that this is where you belong.

Matrix we have in the same way.

The first line, second line, separated by semicolons,

The third line separated by semicolons.

See, wherein 1,2,3, 0, minus 1, 4, minus 2, 1, 0.

The first line B, second line, third line.

A number equivalent to multiplying by a matrix that is multiplied by the number of all elements,

He is now doing in all of this process; doing the right one

process we do not see, giving C.

Here it is collecting B and also, we do not see how it collects,

It gives us as Dr.

B. stands with similar,

B with the beating and giving us as E and F.

They entered our typing attitude of METLAB entry

I gave this to remind me if you said something in Matlab

No need to write it.

Now we have here are,

In the mean multiplying 5'li, with 5 of each item here

multiplying means, as you can see in each case was multiplied by 5.

1 has been hit with 5, which multiplied by 2 to 5, with 3 to 5 have been hit by summer.

And we see the result of this process we saw that means.

D, B, with the gathering.

With elements of collecting the elements of B; 1 0 1 collects writes,

1 2 3 writes the gathering, collecting 3 to minus 1 2 writes.

Like this.

B multiplied by the, you see the multiplication sign

It does not make our hands like a normal x

It shows the impact of this star sign.

A * B you see giving as M,

B * F in the next one to the other and here also gives the general theory

We observe that we see change as a result of the order matrix multiplication.

Whereas in the collection plus B plus B would be equal.

Because after gathering these numbers.

We invest in column B you're getting,

We knock, on the first line of the opposing

We collect the item, and so we find the EUR hit from this definition.

Indeed, we stood 2 0 1 2 We stood with 4,

We stand out from 0 to 3, 4 to bring it here because it also writes 0.

Of course we do not see them, it's already in there power,

We do not need our vision operations, giving the result.

Yet we continue to process,

We know that; A matrix and the B matrix transpose the hit it,

If we take the transpose; The A and B are equal to the product of the coming revolution.

We know that the product that the main theory in reverse order.

We've given are, it was a B..

With B. We stood here.

Now is the inverse of B, we do not see.

Then take the inverse, we do not see.

We wrote here if we wanted; He also gave him give it.

But need we did not want it because it is not required.

This makes the product.

As you can see these two equal.

Equal to the inverse transpose of the product in the product of the coming revolution.

We know that from the theory.

We are done here in providing the desired.

A third embodiment;

The dimensions of matrix B with the difference that the matrix is not the same.

Matrix, see a row of three elements 1, 2, 3.

This is the second line, ie a matrix with two rows and three columns.

4 units in a row, see the BA

There element, 4-column matrix,

but here we see that the three lines.

We now know that, if we want to hit with B.,

We will overthrow the B column, we'll hit him with the number coming equivalent.

In order to perform this operation ...

The number of rows B should be up the column,

compliance and that it should be the matrix of the line up

and B columns so that there must be a matrix.

B. We stood with, so really the number two lines to the lines,

4 columns, we find a matrix B, the number of columns.

But if you look at when you hit the A and B,

We'll get to the column, we will transfer and will hit the B line.

There are two elements in the column.

There are four elements to the B line, this product is not compatible.

The software also makes it distinguished.

This information comes to you so incompatible.

He says it in English.

But it also alerts you if this product.

An important type of problem, solution of equations.

A. X is to get a run in the style of the equation.

A matrix in which a matrix of three three-pointers.

The same problem was solved in the text.

Both in the first and second portions.

X unknown, R on the right side.

We given on the right side.

See where the base is important because it needs to get the r column.

In three rows and three columns of a matrix,

The first line, second line and third line dizeliy them.

So here brings together the overlapping rows of the matrix.

Here also we want to solve; A.

X is equal to r.

So here in disp of the "Display" tab is part of what we call Exhibitions

and as well in these hills in a comma

We say what we want in the future in a letter because she remember.

It gives the output.

Department're doing as follows: A backslash r,

As we've done in the Gaussian elimination when he saw it, the coefficient matrix,

When we unite with the r we find the extended matrix.

The solution is doing, yet we do not see that the Gaussian elimination,

It says it is doing internally and ultimately bring xi.

We want to find a determinant here is because often the determinant

something that was calculated.

det matrix brackets are typed into account the determinants saying.

We ask for this or that; The determinant of a square matrix

There is only one solution is zero.

If zero, or would not be any solution or an infinite number of solutions.

Indeed, as you can see here determinants minus 1 comes in the first instance.

The second and third samples determinant will be zero.

In one of them there will be no solution, there will be endless in one solution.

This Matlab,

It is given to you when you write them;

giving them line by line, we're here because we do not fit in one place.

Matrix writes under equal say.

r is the matrix below, he says he writes.

Det is equal to minus 1, are accounts that we do not do.

So of course you can control by account.

And here's where the resolve to bring the column x writes.

But we also, as we write, the method of Gauss elimination.

So from here on we have learned two things: Det saying braces

When he is going to write her account of what the determinants that account.

You are free to name the left side, you can put what you want.

But you could also say that a number of D

Whether we're also so easy to remind det.

This dete your choice.

Dete in parentheses, which Matlab's own language

but it should be defined in order to make this process of the course.

It also runs the split time, if you notice sign backslash.

You have to write it otherwise can not accept.

Coefficients to be run right here,

here's the solution x in writing and in writing after it here.

And we see that the only solution because we calculate the determinant.

We must do this not just because of our choice to oversee this topic

Or, something I prefer something we have here.

A. In the second example again

X is going to solve r.

We call this summer.

By Gaussian elimination, as it turned out we solve this second example he writes.

but before you write it before you write,

Are you writing r, calculates you say that, you calculate it says.

Then run a command so, r, u, n means to operate said run,

grains that you are running the program and gives the output.

See, here too there is little change from the previous example.

This number 5 6 öbürkü are all the same in the previous example where the same right.

5 minus 3, 10.

Of course, changing a number of determinants to make 0 0

because it is always already made a change to an account number,

It does not bring determinant 0.

[Cough] Just give these printouts.

We write our checks to the determinant of the desire 0

We know ourselves that this equation is now seeing that,

There is no single solution to this equation.

There are infinite or zero solution or solutions.

As you can see here, because there is no solution to the inf

Coming from infinity, infinite, because the last line says it goes like this,

If you look at it you'll see it within that text.

0, 0, 0 is the result of Gaussian elimination.

On the right side is the number one non-0.

0 not sure when this number 0 divides the endless interest.

Take it out when approaching a top forever.

Here the sense of not avaliable, so no sense solution to this report.

Seeing already det determinant calculation was also

if you see it you will understand that the solution also.

You may wonder then why did not he this solution, calculate the determinant

We understand that the sight of the overall solution as the theory taught us.

In the third example again we're solving an equation ax is equal to r.

The coefficient matrix is the same as the previous, but the right side

We have changed a bit, we change the last number.

Here the axis 5 minus 3,

10 is exactly the same, leaving it a challenge of minus 21,

We see that of course this also works with a minus 21 for the previously little studied.

We say fix it again.

On the one hand we say in calculating the determinant.

This is not a requirement, but it's just that we wield our issues under control.

au writes for you, r remember she writes in the future.

Determinant is account, we see that 0.

0 or when we understand that there is no solution or an infinite number of solutions.

It gives a solution as you can see here, the last of 0.

This is one of an infinite solution.

This is a special solution.

If you add any solution in this space you will find a general solution 0.

Do not make it up the way there, but I contend, Nizar ist, I think.

The reason is the following: You will see in the main text, the same sample.

As a result of this last line of Gaussian elimination 0, 0, 0 is happening.

But in terms of the elimination of the right

You find there also 0 expanded matrix.

So going 0 equals 0, no solution at all.

But if you say to this solution is that the endless solution 0

one of x and y, i.e. x can find the first and second variables.

0 if you would give me something else, but not to find another solution.

That gives you a clue that this is very important to find a common solution.

You can find general solutions using it.

If you look at the text, text to both first and second linear

How to understand where the algebra part of the overall solution it.

The presence of the inverse matrix associated with the matrix is an important issue.

Now we have three examples here.

Yet we know that the determinant of matrix If 0 is not vice versa.

0 if there is one, and vice versa.

Now here we define a matrix b.

This B matrix is not random, but in fact you can understand anything.

This is the first example of the matrix of Gaussian elimination.

We say we want to calculate the determinant det,

we have to calculations, we have to use a definition.

What we put into brackets, we calculate its determinant det.

We put in a former au, he has calculated the determinant of a given matrix.

Here we put b matrix just to get a change so what

Putting to show that its accounts.

Here we calculate the determinant b.

As I said before, you can put the name you want to here left.

But can such detb written for the reminder.

Would you put brackets because he det letters in parentheses,

that the powers of MATLAB, in his language.

The Teris describes the inverse of a matrix calculation.

How do you put time into detki b b When calculating the determinant,

When calculating the inverse of the inverse matrix in parentheses.

inverse inverse also comes from the English this word means the opposite.

We do this to ensure the control of a gene product of event.

we multiply b and b opposite should come when the unit matrix,

If done correctly if accounts.

This is a comforting thing, or nothing terms.

This last article we also dispersed data.

This word comes from the display brackets again.

This hill commas important base marks.

He says the letter is not found in English, but we understand.

And after giving the same article.

b b gives after we gave.

We also calculate the b i b i calculates.

We also calculate the determinant, computing the determinant.

It had a small range of changes here writing.

we want to bring before the detb'y detb'y writer here before.

He had us reverse it because we know so relieved that this is not 0.

And b i with b, inversion b, b

You have to go in the opposite when it hits the identity matrix, if done right.

It also stands out, yet we do not see any of them as you can and see

increased unit case matrix.

There are already not be the determinant of the inverse of this matrix 0 0

to assure us that there are opposite.

Now we're getting a matrix as follows.

This was the matrix in the third example of the second example.

We call it calculates the determinant, will be the opposite and let's see

We call calculate the inverse, call accounts and vice versa.

And after this title also writes here.

do not type the brackets are not part of dispersant summer, only this hill

He writes after the comma between the base of the signs.

b writes, b is the determinant, as detb writes, finds 0.

Then we know from theory that the main determinant If 0 no inverse matrix.

Already these things, b reverse in MATLAB

so you do not give a statement in English.

Then you already know that the opposite is also here.

Now let's do a little more for a large matrix.

Of course, when three three-pointers, two hands when able to do so many things duality.

The Quartet also make it possible we can do well with Gaussian elimination

Or, vice versa with cofactors.

Already a simple matrix selected.

See, if you turn a matrix calculation according to the fourth column you need three three-pointers.

Easy, is not a very difficult task.

We matrix here again to give you an example.

See, there are four elements to the four in the rows and rows.

here is threaded successive rows.

Matrix type, writing in our language we understand.

In our view we understand the first line, the second line, third line,

It gives the fourth line.

It accounts for it derteminant.

This is of course a negative one out earlier in an accident.

Minus 1 comes out, that is to say we know there will be the opposite of that matrix.

If the inverse of the matrix inverse word from inv'l inv

We know the meaning of the word.

C I could not be here to remind you what to say but

If you want c t c in the opposite sense.

If you want to Türkçeleştirme thoroughly but it also shows the relationship between the inv'l.

Here comes immediately calculated.

And again, whether in terms of providing the right product,

c c i hit the unit of account have the right if you need to get this inverse matrix.

It gives as you hit D

and as you can see here the unit off matrix.

We can do a little in hand.

Get this example, the first column,

Multiply the first line of the C cycle

2 times 4 times 8 1 minus 5 to minus 5,

I remained 3 we hit the 1-up to -2

1 -2 stayed back, we hit -11'l 0 and 1 output.

After that second line to see çarpsak here 4 times 1 to 4,

0 and will make a contribution of 5, 2 times -2 0 -2,

4, where we had 4 -4 0 even more.

It's happening again this gave 0.

Already machine does not leave them need us to do.

Sheer easy, we're doing this to provide in order to understand the dynamic.

Now the most important in the matrix again

One of the eigenvalues and eigenvectors subject matters.

Eigenvalues and eigenvectors called eigenvalue eigenvector to English.

Extract meaning from German into a special meaning to the word.

Eigen has been accepted impressions here.

eigenvalues of matrix eigen you put into brackets Type

and it gives you the eigenvalues.

Select the row before giving eigenvalues,

of course, it will be more than one eigenvector matrix,

Q writes them as columns of a matrix.

Q is a term we chose here.

Q Why?

Because we adopted in the main text of the Q matrix of eigenvalues.

Lambda again we chose this because a little bit of eigenvalues

a universal representation, lambda 1, lambda 2, lambda We show 3.

This is doing it, we're not here to matrix.

This is the eigenvalues of matrix and take the eigenvalues,

eigenvector gives the columns of a matrix Q.

The eigenvalues of this matrix also diagonalized

matrix gives the numbers on the diagonal.

If you look into this same problem was solved in the second part of this text.

Eigenvalues from 2 to 4, 6 he says.

This order by choosing the little self.

We have the right sort of small to large.

2, 4, 6.

There are 2, 6, 4, so that it no longer has an internal dynamic.

This gives the eigenvalues, eigenvectors of the first

Placing the eigenvalues from column 2 opposite,

The second column 6 corresponding eigenvector,

The third column in the corresponding eigenvector including 4.

Here follows inspections, we can provide.

This matrix Q if it goes zero determinant,

This means that dependent on three columns from each other.

Three different eigenvalues opposed to the exit of each

We know from the general theory is that the independent eigenvectors.

So I got out of determinant 0.

We've also calculate it here.

Calculate detq'y.

-2 Have found calculate detq'y, we also ha completely correct 'call.

Because this is for three discrete eigenvalues different from each other,

three would be independent eigenvector.

Determinant of the matrix formed by them can not be 0.

This determinant is not 0, the Q has vice versa.

Here again (inv) Q Q then find the inverse,

Find the contrary, it is finding.

If you multiply the two units should come in as an inspection.

We also make for call monitoring.

Indeed unit tactics.

Some of them in terms of our training, ourselves,

well we do in terms of our domestic affairs to digest.

Or the machine does not make it wrong.

Coefficients here, looking into your text root 2 divided by 2.

Here, of course, root or stem can not write 2 and 2 divided by 2.

The number, which is the equivalent of writing this.

Of course, the root is not a rational number 2.

So there are infinitely many digits after the decimal point.

And we've chosen to take the four of them.

You could also want to twelve one could want,

You could also want two, yuvarlatarak writer here.

In all of this product because it is also the fact that some rounding

He could be 0,999.

These zero could also be 0.0001.

These would be considered minor rounding will bring.

But this product is provided with the main line.

A second example is; Buddha

A sample was dissolved again in this text, the matrix given.

The first line, second line, third line.

He writes that brought consecutive rows.

We still call it that, eigen (A)

The eigen values, find the eigen vectors.

Eigenvalues it as Lambda,

eigen values written on the diagonal.

Important difference here is the previous example, it is repeated eigenvalues.

When we know from the general theory of eigenvalues repeated,

here two, twice repeated eigenvalues,

it may be from two opposite eigenvalues.

Might as well.

What happens when, when not able check again here.

Because of the matrix Q, which we selected letters,

If you want what we want, you can put the initials of your name,

eigenvector writes the opposing 5 in the first column.

He wrote one of the first to the second pillar from the eigenvalues opposite.

The latter also wrote here.

It corresponds to the same eigenvalues.

Now that we understand that they have been independent of each other linearly dependent Do

We account for determinant.

Because if this column with a previous column depends pulled determinant 0.

Here we see that 0 determinants of acquired accounts.

No matter where the third value.

Where 0 is not important.

So here are three rows, three columns are not going 3 for him.

3 out of such chance.

And here we understand the Q

It is a non-singular matrix.

Therefore, there is also opposite to that determinant is not 0.

We biiy the reverse.

Already we wrote to diagonalization of this matrix.

We write Q -1.

[BOŞ_SES] A'yl to do

We stood and order is important, that you know the main theories.

[BOŞ_SES] Here also

Q Type think it's us

lambda should give the diagonal matrix.

Here was this danger, the same two

When the value could be two independent vectors.

Determinant of interest and that the absence of independent vectors 0

If not Q inv otherwise would not have.

Reverse Q, then we can not do this operation.

But as you can see here, the determinant is different from 0

We are able to reverse this matrix and this matrix diagonalization.

This is one of the main issues we know from the theory.

If we look at this example.

In this example a matrix given as follows; The first line,

The second line, third line, write them on top of, a little bit of a special matrix,

I was not particularly well, which is dissolved in the same text.

We see immediately that the eigenvalues.

This eigenvalues 5pm.

As in the previous example, and one is repeated twice.

Now the main problem 5 had a problem.

5 corresponds to that calculate the eigenvalues found 1.00.

Q writes in the first column.

1 corresponding to the calculated, we found that it wrote.

But there is one more 1 second.

He wrote that calculate the money from him.

Here we see the observed with the naked eye as well.

See Cons of the third row and second row.

If you rotate the axis plus the same number here,

If you turn it the same number minus minus.

Thus two columns of this matrix is not independent.

Therefore, the determinant of this matrix should be 0.

To check this, just the event, in order to understand,

We are going to ensure that the results of the summer.

Here does the opposite for the Q determinant 0.

So here you could want, Q is equal (inverse) Qi said, he would not give you the answer.

And therefore, although here 5, 1,

If this situation does not diagonalizable matrix 1 sorted.

Because we need as we have seen in our previous Q reverse

We hit by AI,

Q'yl to the right again

We will multiply and it's got to be a diagonal matrix.

But the inverse is not Qin Qin reverse we can find here lambda,

matrisleştir not diagonal.

It is written on here but it's just something in terms of their representation.

If it was not for Q inv determinant 0, so the matrix is not diagonalizable.

The same problem as in the case given in the main text should explain a bit more.

Thus we come to the end of the examples we have solved about matlab'l.

My advice to them in comparison with the problems in the main text,

comparing the solution, as you can understand it better as well as the MATLAB

You can even use a assimilate.

See you in another course in the future.

Currently, linear algebra

we finish our problems in the two volumes.

If you watch the first one that got the second one.

I'm his promotion.

Goodbye.