Let S be the rotation of the plane by 45 degree clockwise Gi

Let S be the rotation of the plane by 45 degree clockwise. Give an example of a linear transformation T of the plain, such that T S notequalto S T. (a) Determine the matrices of S, T, T S, S T, and S S. (b) If A and B are the matrix of T and S, verify that (A + B)^2 notequalto A^2 + 2AB + B^2.

Solution

B, the clockwise rotation matrix by 45⁰ is

Cos 45⁰

Sin 45⁰

  - Sin 45⁰

Cos 45⁰

=

1/2

1/2

- 1/2

1/2

Let T be the linear transformation representing the reflection across the X –Axis. Then A =

1

0

0

-1

Then the matrix for the transformation S o T is AB =

1/2

1/2

1/2

-1/2

Also, the matrix for the transformation T o S is BA =

1/2

-1/2

-1/2

1/2

Since AB BA, therefore S o T To S.

The matrix for the transformation S o S is BB =

0

1

-1

0

(b) A+B =

1+1/2

1/2

-1/2

-1+1/2

So that (A+B)² =

1+2

1

-1

1-2

Also , A² =

1

0

0

1

2AB =

2

2

2

-2

B² =

0

1

-1

0

So that A² +2AB +B² =

1+2

1+2

-1+2

1-2

It may be observed that (A+B)² A² +2AB +B²

Cos 450	Sin 450
- Sin 450	Cos 450