代写Math 2568、代做orthogonal vectors、代写Java，c/c++编程、Python程序语言调试

日期：2019-04-28 08:47

Math 2568 Practice Final April 16, 2019

Problem 1. True or false? Explain your reasoning. If false, find a condition on A which makes it

true.

(1) If the columns of A are linearly independent, then so are the rows.

(2) If A is invertible, then so is AT

.

(3) If Ax = b has a unique solution for some b ∈ R

m, then nullity(A) = 0.

(4) If the only eigenvalue of A is 1, then A is diagonalizable.

(5) If A maps orthogonal vectors to orthogonal vectors, then A is orthogonal.

Problem 2. Short answer. The answers to all questions below are a single digit between 0 and 9.

(1) Compute the determinant of.

(2) Suppose 1 is an eigenvalue of A. What number is then an eigenvalue of A + I?

(3) Suppose A is a 7 × 9 matrix whose rank is 4. What is the dimension of the row space of A?

(4) The matrix

λ 1 0 0

0 λ 0 0

0 0 λ 0

0 0 1 λ

has only one eigenvalue. What is its geometric multiplicity?

(5) Suppose

1 2 1

0 1 0

1 5 k is not invertible. What is k?

Problem 3. Find the Jordan normal form of the following matrices:

Problem 4. Find an orthogonal matrix Q such that QT AQ is diagonal for

Problem 5. Consider the matrix

(1) Find 3 linearly independent solutions X1(t), X2(t), X3(t) of the differential equation X0

(t) =AX(t).

(2) Find a particular solution of the form a1X1(t) + a2X2(t) + a3X3(t) when X(0) =.

Problem 6. Find bases for the column space, row space, and null space of the matrix

.

1

Problem 7. Let v ∈ R3 and let A be a 3 × 3 matrix. Suppose v, Av, A2v are all nonzero and

A3 = 0.

(1) Show that B = {v, Av, A1v} is linearly independent. Deduce B is a basis for R3.

(2) Calculate [LA]B where LA is the linear map given by LAx = Ax.

Problem 8. Suppose A, B are n × n matrices such that AB = In. Show that BA = In.

2