408 CHAPTER 18. LINEAR FUNCTIONS

48. Let A be an n × n matrix and let p(λ ) be the minimal polynomial of the aboveproblem. By the fundamental theorem of algebra, this can be factored as

m

∏i=1

(λ −µ i)

where µ i ∈C. Thus, from the above problem, ∏mi=1 (A−µ iI) = 0. Explain why there

is a vector vk such that uk ≡ ∏i̸=k (A−µ iI)vk ̸= 0. Explain why (A−µkI)uk = 0.Thus A has an eigenvector for each of the µ i. Note that you must allow all arithmeticto take place in C because the eigenvalues µ i are only known to be complex numbers.

49. Let θ ∈R. For x a vector in Rp, p > 1, let Tθ be defined as follows. Place x with itstail at the origin and rotate through an angle of θ . If θ > 0, rotate counter clockwiseand if θ < 0 rotate clockwise as in trigonometry. Argue with elementary geometrythat Tθ is a linear transformation. In case p = 2, explain why the matrix of Tθ , calleda rotation matrix, is (

cosθ −sinθ

sinθ cosθ

)It amounts to justifying the following picture.

e1

e2

θθ

(cos(θ),sin(θ))(−sin(θ),cos(θ)) T (e1)

T (e2)

50. Now note that Tθ Tα = Tθ+α . Using matrix multiplication and the above problem,derive with virtually no effort the formulas for sin(θ +α) and cos(θ +α).