Let T be the linear transformation T: R² R² that takes aPoint (x,y) ER² and reflects it across the x-axis, then rotatesit counter Clockwise about the origin through an angleof 21 1².24i. Let A be the standard matrix for T. Compute det (A²).ii Is T one-to-one? Is Tonto? Explain.

By reflection of a point (x ,y) across x -axis the x -coordinate remains the same and the y coordinate becomes its negative. If a point (x , y ) makes an angle θ with the positive x - axis and it is rotated about the origin through an angle α then the new point (x1 , y1 ) is x1 = r cos (θ+α) y1 = r sin (θ+α) where r = x2+ y2 First we find the standard matrix for T The standard basis of R2 is {(1,0) , (0,1)} (1,0) after reflected across x - axis is ( 1, 0) itself. When (1,0) rotated anticlockwise by an angle of 2π/3= 120° θ=0 (the angle made by (1,0) with positive x -axis) and α =120 r = x2+ y2 = 12+ 02 = 1 The new point is ( 1 cos 120 , 1 sin 120 ) = ( - 1/2, 3 /2 ) (0,1) reflected across x - axis is (0 ,-1) θ = -90 , α = 120 θ+α= 30 and r = 1 The new point is ( cos 30 , sin 30) =(3 / 2 ,1/2) The standard matrix is A =-1/23/23/21/2 A2 =-1/23/23/21/2 -1/23/23/21/2 = 1001 = I , the identity matrix. Det( A2) = Det (I) = 1 "If T is a linear transformation from a vector space V to V itself then the matrix representing T is a square matrix. T is invertible if determinant of the corresponding matrix is non- zero." Using the above fact since det (A) is non zero the linear transformation T is invertible. So T is both one-one and onto