A reflection in R2 across a line L through the origin is a l

A reflection in R2 across a line L through the origin is a linear transformation. Suppose that M is the standard matrix for reflection across L.

a. What happens if you apply the reflection transformation twice?

c. Show that it follows that M^?1 = M .

d. Use matrix algebra to show that P^2 = P .

e. What does this tell you about applying the projection transformation twice?

f. Use the equation P^2 = P to show that P is not invertible. [Hint: What would happen if you multiplied both sides by P^?1 ?]

Many thanks for your help!

b. Explain why this tells you that M2 -12 must be true

Solution

(a).When we apply the reflection tranformation (representing reflection across a line L through the origin), the original position of a point/vector gets restored.

(b) Therefore, applying the reflection tranformation twice is the same as multiplication by the identity matrix I₂.Thus. MM = M² = I_2.

(c ) Hence M^-1(MM) = M^-1I₂ or, (M^-1M) M = M^-1 or, M = M^-1.

(d ) If P = 1/2(M +I₂) , then P² =P.P= [1/2(M+I₂)].[1/2(M+I₂)] = 1/4(M²+M+M+I₂) =1/4(2M+2I₂) ( as M² = I₂) = 1/2(M+I₂) = P.

(e). P² = P, implies that applying theprojection transformation twice has the same effect as applying it only once.

(f) P² = P. Let us assume that P is invertible. Then on multiplying both sides by P^-1 , we have P^-1P² = P^-1P or, ( P^-1P)P = I₂ or, I₂P = I₂ or, P = I₂ which is a contradiction. Hence P is not invertible.