Have A Nice Winter Break!

Linear Algebra II, FINAL, Yung-fu Fang, 2011/01/10 Show All Work

1. (a) State some sufficient and necessary conditions for A or T being diagonalizable. 3% each

(b) State the definition of Pseudoinverse. [5%]

(c) State the Singular Value Decomposition. [5%]

(d) State the definition of a vector space. [5%]

(e) State the definition of an inner product. [5%]

(f) State the Cayley-Hamilton Theorem and the definition of minimal polynomial. [5%]

2. Let {u1, u₂, u₃} be a set of linearly independent vectors in an inner product space. Use the Gram-Schmidt Process to compute the orthogonal vectors{v1, v₂, v₃}. Then normalize these vectors. [10%]

3. Let T : R² → R² be the rotation by θ. Prove that T is a linear operator. Is T diagonalizable?

Explain! [10%]

4. Let T be a normal operator on a finite-dimensional complex inner product space V . Use the spectral decomposition T = λ₁T₁+ λ₂T₂+· · · + λkT_k to prove that there exists a normal operator U

on V such that U² = T . [10%]

5. Let V and W be finite-dimensional inner product spaces. Let T : V → W and U : W → V be linear transformations such that T U T = T , U T U = U , and both U T and T U are self-adjoint. Prove

that U = T^† [10%]

6. Find the singular value decomposition and A^† for A =

( 1 1 −1 1 1 −1

)

. [10%]

7. Let A∈ M3×3 and diagonalizable with distinct eigenvalues λ₁, λ₂, λ₃ and eigenvectors v₁, v₂, v₃ Find A = QDQ⁻¹ and A = U ΣV^∗. Express the matrices Q, D, U , Σ, and V in terms of λ₁, λ₂, λ₃ and

v₁, v₂, v₃. [10%]

8. Let A be a 2× 2 matrix. Let λ ∈ R and ξ ∈ R² such that (A− λI)ξ = 0. Suppose that the null space of A− λI is 1-dimensional and (A − λI)² is a zero matrix. Show there is a vector η ∈ R² such that (A− λI)η = ξ. Show that β = {ξ, η} is an ordered basis for R². Find [L_A]_β. Find the matrices

Q and J such that A = QJ Q⁻¹. [10%]

9. Let A =





3 1 −2

−1 0 5

−1 −1 4



. Find a Jordan Canonical Form for A and the minimal polynomial

of A [10%]

Have A Nice Winter Break!