And at first for the exercise five, we have a standard basis that for, we have the standard

base that EX0, EY0, and EZ0. And then we, following we use the Einstein's

Summ Convention, and then we use that double indexes over, always over the X, Y, and Z.

And then we have the two matrix, the first matrix is A, it's equal to AIJ, EY0, and this

is the A matrix, and there's the X matrix.

So we have these three matrix, and each index will always go over for the X, Y, and Z.

And then first we need to calculate the A is that for the transpose of X transpose and

Y, and at first we need to know what is X. And for the X, because we go over for the,

go over X, Y, and Z, so for the X it will be equal to X, X, EX0, X, Y, EY0, and XZ,

EZ0. And for the EX0, it's 1, 0, 0. And for the EY0, it's 0, 1, 0. And for EZ0, it's

0, 0, 1. It's in the fixed space coordinate system, the normal system, the XYZ. And so

for the X vector, it will be equal to X, X, XY, and XZ. This is the X vector. And so similar

for the Y vector, it will be equal to YX, YY, YZ. It's similar as here. It's YX, YY,

and YZ. So I know we can calculate this X transpose with Y, and we can have the X transpose.

This one will be equal to, this transpose is equal to X, X, XY, and XZ. And with this

one, it will be YX, YY, and YZ. And this one will be X, X, YX, X, Y, Y, Y, Y, Y, Y,

Y, Y. So it will be equal to this one. And it can also write in this form that XI, YI.

This is for the scalar product. And then for the tensor product, second one is for the

tensor product B. For the second task, that was the tensor product. It's this one, X,

it will be equal to X divided by Y transpose. And here, we can write this as X, because

X equal to here, XI, EI, zero. Hello? Yeah, XI, EI, zero. And then for this Y transpose,

this tensor product, that also that YI, YG, EG zero transpose. Like this one. And then

we can give that to XI, YG, EI zero, EG zero transpose. And this EI zero times the dot,

dot times the EG zero transpose. And it will like this form. So X, it will equal to XI,

YG, EI zero. Because here, this matrix, it's three times one. Three times one matrix, vector.

And this transpose, it's one times three. So totally here, then come here, it will be

a three times three matrix. So I will write the form at here.

X, X, X, X, X, X, X, Y, Y, X, X, Y, Y, Y, Y, Y.

So because we use the Einstein's Stump convention,

so from here it's the real state, it will be like this one.

This three times three matrix.

So this is that tensor product.

For X tensor product Y,

it will be like this one.

Then we need to calculate the matrix,

the A times X.

We will calculate this matrix, the A times X.

And from here the A matrix,

so we need to know what is the A matrix.

And from A matrix, we have the,

it's already given at here.

And here is also tensor product.

And because here is tensor product,

so it also means that this one will be a three times three matrix,

like this one.

So A matrix, it will be like this.

So this will be that A matrix.

This one.

And so if we want to know the A matrix times X,

here we will use that.