Reference no: EM132495760

Question 1: Let V = M_2x2(C) be the vector space over the field C of 2 x 2 matrices with entries in C, and let W = P₃(C) be the vector space over the field C of polynomials of degree at most 3 with coefficients in C.

Consider the following bases B and C, of V and W, respectively:

Let T: V →W be a function defined as follows:

(a). Show that T is a linear transformation.

(b). Find the matrix A of T with respect to the bases B and G. Show your work and briefly explain your process.

Q1(c). Suppose D is a mysfery basis for W, and  .

Write an expression for the vector of coordinates Φ_D(p(t)) in terms of Φ_B, the matrix A of T from (b), and the change-of-basis matrix M_C→D.

Your expression should tell us how we could use fi to directly compute Φ_D(p(t)) we were explicitly given the matrix M_C→D and the coordinate vector on Φ_B. 

Question 2. Consider the functions f:M_mxn(C)→ R_≥0 and g: M_mxn (C) → R_≥0 defined as follows:

f(A) = ∑_i=1^m∑_j=1ⁿ|A_ij|

g(A) = max{|A_ij|: 1 ≤ i ≤m, 1 ≤ j ≤ n}

(a). Prove that f is a norm for M_mxn(C) (using the definition of norm).

(b). Prove that g is a norm for M_mxn(C) (using the definition of norm).

(c). For each of the following norms on M_2x2 (C), compute the norm of the matrix

Show your work.
Q2(c)-i The norm f.

Q2(c)-ii The norm g.

Q2(c)-iii The matrix 1-norm ||.||₁.

Q2(c)-iv The matrix ∞-norm ||.||_∞.

Q2(c)-v The matrix 2-norm ||.||₂.

Question 3. Let ,where x ∈ R and x > 0.

(a). For which value(s) of x do we have |||RA^-1||_∞ ≥ 1 and || A^-1R||_∞ < 1 ?

(b). For the value(s) of x found in (a) and the ∞-norm, what (if anything) does Theorem 16.1 (or Theorem 12.5 in Mike Newman's notes) guarantee? If applicable, give the bounds on ||(A+R)^-1||_∞. Briefly explain your reasoning.

Question 4. You are given that

Suppose R is a 4 x 4 matrix whose entries are all real numbers that lie on the open interval (-1/24,1/24).

(a). Find the 1-norms of A and A^-1 and give an upper bound on the 1-norm of R.
(b). Find the ∞-norms of A and A^-1 and give an upper bound on the ∞-norm of R.

(c). Based on the 1-norms of these matrices, can you conclude that A + R is invertible? Explain.

(d). Based on the ∞-norms of these matrices, can you conclude that A+R is invertible?
Explain.

(e). Is A+R is invertible? Explain.

Question  5(a). Give an example of a 2 x 2 matrix A ∈ M_2x2 (C) such that A has at least one non-real entry but all the eigenvalues of A are real. Briefly justify your answer.

(b). Give an example of a 2 x 2 matrix B ∈ M_2x2 (R) such that B has all real entries but all the eigenvalues of B are non-real complex numbers. Briefly justify your answer.

(c). Give an example of a 2 x 2 matrix C ∈ M_2x2 (R) such that C is not diagonalizable. Briefly justify your answer.

(d). Give an example of a 2 x 2 matrix D ∈ M_2x2 (R) such that D is unitary and none of the eigenvalues of D are equal to ±1. Briefly justify your answer.

Question 6. Let A be an n x n matrix with entries in C.

For each column of A, define a column-based radius c_j: = ∑_i≠j|A_ij|, and a circle in the complex plane given by the inequality |z - A_jj| ≤ c_j.

Prove that each eigenvalue of lies in one of these circles, |z - A_tt) ≤ c_t, for some such that 1 ≤ t ≤ n.

hints: You may use the fact that A and A^T have the same eigenvalues and modify oppropriately the proof of the Gershgorin Circle Theorem seen in class.

Question 7. Let

(a). Compute the Gershgorin circles for A using the row-based radii. Draw a picture of the circles.

(b). Compute the Gershgorin circles for A using the column-based radii. Draw a picture of the circles.

(c). Based only on the Gershgorin Circle Theorem (using either the row- or column- based radii) what (if anything) can you conclude for each of the following questions? Be as specific as possible. Your answers should only state conclusions that can be made from the Gershgorin Circle Theorem stated in class (Theorem 17.3 in LEC 17 notes or Theorem 15.1 in Mike Newman's notes or the column version of these theorems). Your answers should not be based on explicitly solving for the eigenvalues of N, explicitly computing its inverse, etc. Draw your conclusions only from The Gershgorin Circle Theorem.

(c)-i. What is the minimum number of distinct eigenvalues that A has (based on Gershgorin Circles)? Briefly explain.

(c)-ii. Can you determine the algebraic multiplicity of any of these eigenvalues (based on Gershgorin Circles)? Briefly explain.

(c)-iii. Can you conclude whether A is invertible (based on Gershgorin Circles)? Briefly ex- plain.

(c)-iv. Can you conclude whether A is diagonalizable (based on Gershgorin Circles)? Briefly explain.

Question 8. Let

(a). Find the characteristic polynomial of A. Show your work.

(b). For each eigenvalue λ of A, determine its algebraic multiplicity mult(λ) and its geometric multiplicity dim(λ). Compute a basis for each eigenspace.

(c). Give the Jordan form of A and briefly explain how you were able to determine the sizes of all the Jordan blocks.

(d). Can you find an invertible matrix P such that A = PJP^-1 where J is the Jordan form of A? If you think so, then beware: starting with an eigenvector in the eigenspace of some eigenvalue λ, you may need to solve for a generalized eigenvector; however, not every eigenvector in the eigenspace will produce solutions for the generalized eigenvectors. If you carefully choose an eigenvector from within the eigenspace, then you will be able to solve for the generalized eigenvector(s). Show your work and explain your process. Reasonable and justified attempts at findingP may be awarded 1 or 2 bonus points.

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