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So as we have seen previously, an emperor can be seen as a victor or as matrix.

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So let us discuss a bit what we can do with these areas in terms of multiplying vectors, for example.

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So here I have written down two new vectors Vector one and Vector two, and they just contain some random

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numbers.

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So let us around the cell.

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And now what we can do is we can use the functions that are built into python or better to say, into

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Nampai, to calculate the Dot product and also with the cross product of these two vectors.

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So the DOT product is defined as and p dot.

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And then we typed Yochi.

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And if you dot of vector one comma factor two and the result is a nine.

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So in case you don't know how a dot product is calculated.

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Yeah, basically, you multiply these two numbers, plus the product of these two numbers, plus the

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product of these two numbers.

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So we have two plus three, which is five plus four is nine.

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Then we can also calculate the vector product or so-called cross product by typing NPR Dot Cross.

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Once again, we used a two arguments Vector one and Vector two, and the result will be an array.

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So a vector with a components one minus two and one.

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So I think you will notice, but a vector product does not commute.

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So this means the order in which we provide these arguments does matter for the results.

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So when I change this year to a vector to come, our vector one gives me a different result and gives

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me the opposite sign.

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So this means, of course, we will learn later how to define functions.

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And of course, you could say I want to define my own cross product function.

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Or you could also say, I want to define no function at all and just calculate all the components every

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time I use across product.

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That's OK.

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But that, of course, takes a bit more time than just using the command that is provided by.

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And also one more comment.

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Most of these non-player routines, they rely on other programming languages, and they are really optimized.

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So in terms of calculation time, so they are really, really fast to calculate and much faster than

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what we would do it ourselves.

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So another typical thing that you want to do with vectors is calculating their norm.

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So basically their length.

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So the command here is an p dot.

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And no, it's not just norm, but we first need to say it arc for linear algebra and then norm.

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So I cannot really tell you why this is the case, but this is what the command is and p fill in OG

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and more.

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And for example, for vector number two, the norm is some floating point number five point something.

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So as I mentioned, we don't have to use this function.

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We can also do it ourselves.

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So we could say the norm is calculated as the square root and p square root of the X component squared.

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So this would be Vector two and then in square brackets zero and the square would be this.

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And then we add the square of the Y component and the square of the Z component.

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And the result is exactly the same as when we were using the function that was provided by Nampai.

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So what else can we do?

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Well, we can also use another Nampai function, which is the and p some function, because we could

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just say, since our vector is an array, we can do component wise.

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Yeah, mathematical operations.

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So when I write factor two to the power of two, the result will be that every component of our vector.

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So this would be two three four will be squared.

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So the result will be an array with the components for nine and 16.

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So now what we have to do is we have to write and p got some.

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So this will add up these three numbers.

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So this would be exactly this one.

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And know we only have to write NPR Dot Square Root and we have our norm.

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So you see three different methods to calculate the norm.

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But of course, the most easy one is to just use to come on that is provided by Nampai and the Lindborg

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module.

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So that's all I wanted to say about vectors.

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Of course, people work with vectors in the following course and you will learn more about this.

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But these are just like the really the basics.

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So now let's continue and let's define some matrices.

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So as we have learned, a matrix is just a multidimensional array.

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So we write Mporei and the science is wrong, the normal brackets, and then we make one square brackets

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and then another one.

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And here we write the first line.

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So I just take some random numbers.

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Yeah, minus four or five and minus six.

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And another one, I don't know, seven, minus eight nine.

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And let's define also another matrix, which we could, for example, make diagonal or also.

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Yeah, maybe we do it like this one one one zero two two and zero zero three.

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So just some random numbers doesn't really matter so much.

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So and also we have seen that once we write, once we call these functions or of these functions, but

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these variables, then it gives us the output in such a matrix form, which is really nice.

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So what we can do with matrices is, of course, we can multiply them so we can write Matrix one times

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Matrix two.

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However, as we have learned, the and pitot array function, when you apply a mathematical operation,

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the operation will be applied to every component and the similar thing happens here.

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So when we look at our two matrices, you can see that the result here is just a component wise multiplication.

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So one times one is one minus two times one minus two.

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For example, here, five times two is 10.

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And this is not really how you multiply matrices.

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So this expression here is not really reasonable and not really useful.

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What we want instead is a matrix multiplication and the correct command.

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Here is and P dot match model for matrix multiplication.

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And then two arguments would be matrix one.

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And of course, Matrix two.

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This time we get the correct term.

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For example, the central number here would be the the dot product of these of this vector with this

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vector.

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So it would be minus four times one, which is minus four plus five times two, which is 10 and then

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plus minus six times zero, which is zero.

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So we have minus four plus 10, which is, of course, six.

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So you see, it really is a matrix multiplication.

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And we can also do is we can reverse the arguments and say, calculate the product of Matrix two and

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one.

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And of course, the result will be different because like the cross product for vectors, the matrix

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multiplication is not computing for the matrices.

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So another very important thing for matrices is calculating the determinant and also the inverse, which

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so these two operations are kind of related.

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So here we just use two commands and p dot linear algebra dot determinant Matrix two, for example,

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which has a matrix sorry, which has a determinant of six.

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And then we can also write and p dot linear algebra dot inverse, which is Matrix two or yeah, where

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we used to function on Matrix two.

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So what's the property of an inverse?

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Well, if you calculate the Matrix product so and p dot matrix multiplication of this matrix and the

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matrix itself, it gives us the one matrix.

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So just ones on the main diagonal.

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All right.

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So you see, we have calculated the the inverse, the determinant, and we have checked everything is

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correct.

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And of course, just a mathematical fact you will propose and notice.

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So if we calculate the inverse of Matrix one, we see something strange happened to you.

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We get very, very.

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Large numbers here for two to power so far, times 10 to the power of 15, so this doesn't seem reasonable.

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What is happening here?

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Well, the thing is the determinant of this number is zero.

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So there's just zero, basically.

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So if a determinant of a matrix is zero, then you cannot calculate the inverse, which gives us this

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problem here.

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So sometimes you have to be a bit careful because the correct mathematical output would have been the

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inverse of this matrix doesn't exist.

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But of course, since it's a computer that's a bit hard to do.

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So the result will just be some unreasonable number.

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So be a bit careful when you use these commands.

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But overall, working with matrices and vectors is pretty easy and pretty straightforward.

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You just have to be careful that all the ranks of fitting and sometimes you have to even transpose the

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matrices.

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And yeah, so that's as you note from mathematics so that this matrix multiplication works.