Please help with 3c. Thank you! (3) In this problem, you will show that an 2 × 2

Question

Please help with 3c. Thank you!

(3) In this problem, you will show that an 2 × 2 orthogon matrix CT must describe a rotation or a reflection in the plane that, either b =-c and d = a, or b c and d =-a. W n each case (b) Show t et(A)- 1, then there is an angle that sin(9) - sin(0) cos( The associated transformation is clockwise rotation by an (c) Show that, if det(A) =-1, then there is a vector v such that, for all w in R2 Note: The associated transformation is reflection across the line perpendicular to v Hint: The vector v will be an eigenvector with eigenvalue-1

Explanation / Answer

Ans:3-

The simplest orthogonal matrices are the 1 × 1 matrices [1] and [1] which we can interpret as the identity and a reflection of the real line across the origin.

The 2x2 matrix have the form:

A= a b

c d

which orthogonality demands satisfy the three equations

1 = a2 + b2 ,

1 = c2 + d2 ,

0 = ac + bd

In consideration of the first equation, without loss of generality let a = cos , b = sin ; then either

b = c, d = a    or b = c, d = a.

We can interpret the first case as a rotation by (where = 0 is the identity),

and the second as a reflection across a line at an angle of /2.

then

Rotation is :

   cos Thete - sin Theta

   sin Theta cos Theta

and

Reflection is:

cos Theta sin Theta

   sin Theta - cos Theta

The special case of the reflection matrix with = 90° generates a reflection about the line at 45° given by y = x and therefore exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):

0 1

1 0

The identity is also a permutation matrix.

A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.

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