problem 4 Consider a composite transformation from R2 to R c

problem 4

Consider a composite transformation from R2 to R clockwise rotation Tby 30 followed by 4. reflection S in the line x y 0. Give standard matrices for the transformations Tand S. (b) Use these two matrices to find the standard matrix of the composite transformation.

Solution

(a). In R², the matrix for clockwise rotation by an angle of 30⁰ is M =

Cos 30⁰

Sin 30⁰

-Sin 30⁰

Cos 30⁰

  

i.e. M =

(3)/2

1/2

-1/2

(3)/2

Also, in R² the matrix for reflection in the line x+y = 0 or y = -x is N =

1-(-1)²

2*(-1)

2*(-1)

(-1)²-1

i.e. N =

0

-2

-2

0

M and N are the standard matrices for T and S respectively.

(b) The standard matrix for ToS is MN =

-1

-3

-3

1

  The standard matrix for SoT is NM =

1

-3

-3

-1

Cos 300	Sin 300
-Sin 300	Cos 300