Question

Let a, b eR, not both zero, and for A-(ab let LA : R2 → R2 be given by La@)=( b a)(z2 Show that geometrically, La is a rotation of the plane, composed with a dilation of the plane. What is the angle of rotation? What is the dilation factor?

Answer 1

Let A = (a b - b a) where a, b elementof IR, not both zero, now, A = squareroot a^2 + b^2 (a/squareroot a^2 + b^2 b/squareroot^2 + b^2 - b/squareroot a^2 + b^2 a/squareroot a^2 + b^2) note a^2 + b^2 greaterthanorequalto 0 as a, b elementof IR a^2 + b^2 > 0 not both a, b can be 0 i.e squareroot a^2 + b^2 is well defined = (squareroot a^2 + b^2 0 0 squareroot a^2 + b^2) (a/squareroot a^2 + b^2 b/squareroot a^2 + b^2 - b/squareroot a^2 + b^2 a/squareroot a^2 + b^2) Hence L_A (x) = (squareroot a^2 + b^2 squareroot a^2 + b^2) (a/squareroot a^2 + b^2 b/squareroot a^2 + b^2 - b/squareroot a^2 + b^2 a/squareroot a^2 + b^2) (x_1 x_2) note that theta = sin^- (b/squareroot a^2 + b^2) , we can rite L_A as L_A = D_A R_A where R_A (x_1 x_2) = (cos theta sin theta - sin theta cos theta) (x_1 x_2) Angle of rotation sin^- 1 (b/squareroot