讲解：Math 2568、orthogonal vectors、

Math 2568 Practice Final April 16, 2019Problem 1. True or false? Explain your reasoning. If false, find a condition on A which makes ittrue.(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 ∈ Rm, 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 00 λ 0 00 0 λ 00 0 1 λ�has only one eigenvalue. What is its geo代写Math 2568作业、代做orthogonal vectors作业、代写Java，c/c++编程作业、Pythonmetric multiplicity?(5) Suppose�1 2 10 1 01 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 forProblem 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.1Problem 7. Let v ∈ R3 and let A be a 3 × 3 matrix. Suppose v, Av, A2v are all nonzero andA3 = 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转自：http://www.7daixie.com/2019042913633738.html