Question

(12) (after 3.3)

(a) Find a linear transformation T : R2 → R2 such that T (x) = Ax that reflects a

vector (x1, x2) about the x2-axis.

(b) Find a linear transformation S : R2 → R2 such that T(x) = Bx that rotates a

vector (x1, x2) counterclockwise by 135 degrees.

(c) Find a linear transformation (with domain and codomain) that has the effect

of first reflecting as in (a) and then rotating as in (b). Give the matrix of this

transformation explicitly. How is this transformation related to T and S?

(d) Find the matrix representing the linear transformation that first rotates as in (b), then reflects as in (a), and then rotates backwards (i.e., clockwise by 135

degrees).

(e) What matrix do you get if you repeat the sequence in part (d) ten times?

Write this matrix in terms of A and B. Can you write this matrix explicitly?

Answer 1