9. Let A =. Compute for any positive integer n.

(a) Find possible eigenvalues of A.

(b) Determine the rank of A.

7. Prove that if Q is an orthogonal matrix, then det(Q) = 1.

8. Let V and W be two vector spaces and let T : V → W denote a linear transformation. Suppose that N(T) and R(T) are null space and range of T, respectively. Show that N(T) and R(T) are subspaces of V and W, respectively.

    Notation: R is the set of real numbers, and C is the set of complex nunbers. If F=R or C, denote by (F) the n n matrices with entries in F. If A ∈ (F),denote by ∈ (F) the transpose of A. Denote by the n x n identity matrix and the n x n zero matrix. Problem 1 :Let i = √-1∈ C be a root of X² + 1. Let                   v₁ = (1,0,-1), v₂ = (1 + i,1 − i, 1), v₃ = (i, i, 1).Show that {v₁, v₂, v₃} is a basis of C³ and express the vector v₄ = (1,0,1) as a linear combination of v₁, v₂ and v₃, namely find α₁, α₂, α₃ ∈ C such that v₄ = α₁v₁ + α₂v₂ + α₃v₃.

Problem 2 :Let v₁ = (0,3,3, 1), v₂ = (2, 1, -3, 7), v₃ = (1, 8, 6, 6), v₄ = (1, 10, -4,2) be vectors in R⁴. Let W₁ = span{v₁, v₂} and let W₂ = span{v₃, v₄}. Find the dimension and a basis of W₁∩W₂.

(1) Find an orthogonal matrix P ∈ M₃(R) such that AP is a diagonal matrix.

(2) Find the singular value decomposition of A. In other words, factorize A = , where U ∈ M₂(R) and V ∈ M₃(R) are orthogonal matrices and Σ ∈ (R) is of the form Σ = , λ₁ ≥ λ₂ ≥ 0

(1)  Find the dimension of Ker T.

(2) Show that T is diagonalizable.