# Linear transformations!

• Apr 19th 2011, 04:00 AM
mechaniac
Linear transformations!
http://quicklatex.com/cache3/ql_efc8...e92734e_l3.png

need some help getting started. I know how to project, rotate, reflect.
But i´m stuck anyways :)

Thanks!
• Apr 19th 2011, 04:32 AM
FernandoRevilla
Perhaps you meant v -> v x u instead of u -> v x u . If so, find first

v x u = ( v_1 , v_2 , v_3 ) x ( -3 , 2 , 3 ) = ...

and apply the reflection to v x u .
• Apr 19th 2011, 07:43 AM
mechaniac
oh, its v=(-3,2,3) not u.
• Apr 20th 2011, 05:43 AM
mechaniac
Thanks fernando for the help.

So i did it this way:

http://quicklatex.com/cache3/ql_a0ac...19bddb6_l3.png

But its not the correct answer. What did i do wrong?
• Apr 20th 2011, 07:19 AM
FernandoRevilla
Quote:

Originally Posted by mechaniac
But its not the correct answer. What did i do wrong?

If u = ( x , y , z ) then

T_1 ( x , y . z ) = ( -3 , 2 , 3 ) x ( x , y , z ) =

... = ( 2 z - 3 y , 3 z + 3 x , - 3 y - 2 x )
• Apr 20th 2011, 08:07 AM
mechaniac
oh i did a misstake with the crossproduct. There is a typo in the T2 matrix to but no worries, got it now! Big thanks :)
• Apr 20th 2011, 02:33 PM
HallsofIvy
The simplest way to find a matrix representation of a linear transformation is to look at what the transformation does to the basis vectors. For example, to map <1, 0, 0> onto <1, 0, 0> X <-3, 2, 3> means to map <1, 0, 0> onto <0, -3, -3>. To map <0, 1, 0> onto <0, 1, 0> X <-3, 2, 3> means to map <0, 1, 0> onto <3, 0, 3>. To map <0, 0, 1> onto <0, 0, 1> X <-3, 2, 3> means to map <0, 0, 1> onto <-2, -3, 0>. The matrix giving that is
[ 0 3 -2]
[ -3 0 -3]
[ -3 3 0]
the matrix having those results as columns. To see that, mutiply that matrix by <1, 0, 0>, <0, 1, 0>, and <0, 0, 1> succesivly.

Reflection in x= z maps <1,0, 0> into <0, 0, 1> and <0, 0, 1> into <1, 0, 0>. Since <0, 1, 0> is in that plane, it is mapped into itself.

The matrix giving that transformation is
[ 0 0 1]
[0 1 0]
[1 0 0]

To find the matrix representing both of those transformations, multiply them (in the proper order, of course).