- #1

Hannisch

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## Homework Statement

Determine the matrix A for the linear map T:

**R**

^{3}→

**R**

^{3}which is defined by that the vector

**u**first is mapped on

**v**×

**u**, where

**v**=(-9,2,9) and then reflected in the plane x=z (positively oriented ON-system). Also determine the determinant for A.

## Homework Equations

## The Attempt at a Solution

I actually started with the determinant and said that since the first mapping is a projection the determinant of that is =0 -- thus, the determinant for the whole thing is 0, since det(B*C)=det(B)*det(C) and in this case, A=S*P, where S is the determinant for the reflection (which is what we usually use in Swedish) and P is the projection.

Anyway, I started with S (for practise, if nothing else):

The plane will have the equation x-z=0 in its normal form and thus the normal to the plane is <1,0,-1>. So if I call vector

**w**<a,b,c>, the reflection in the plane is:

<a,b,c> + t<1,0,-1> = <a+t,b,c-t>.

<a,b,c> + (t/2)<1,0,-1> needs to be on the plane and thus the coordinates for that vector need to fulfill the plane equation,

(a+t/2) - (c-t/2) = 0, t= -(a-c).

<a-(a-c),b,c+(a-c)>=<c,b,a>.

Thus, the reflection matrix is:

[tex]S = \left( \begin{array}{ccc}

0 & 0 & 1 \\

0 & 1 & 0 \\

1 & 0 & 0 \end{array} \right)\]

[/tex]

I didn't think that was too difficult, but now I'm entering the confusing part.

I can quite easily calculate

**v**×

**u**, if

**u**=<x,y,z>.

(

**v**×

**u**= < -9y+2z, 9x+9z, -2x - 9y>.)

But (and I was thinking this from the very beginning)

**v**×

**u**is orthogonal to

**u**(by definition of the cross product, because it creats a vector orthogonal to the plane containing

**v**and

**u**). But there can't be any projection if it's orthogonal, right? So I thought that if they all stay the same it should be the identity matrix and thus S*I=S, but that is not correct. And now I've got no idea what to do... And I didn't really want it to be the identity matrix, because that would not mean that the determinant is 0.

The conclusion is that I'm very confused.