Math 232 Homework Assignment 05

- Due 4:30 PM Monday, February 21st 2000

• Section 3.2, pages 202-203:
  • 4, 8, 12, 16, 18, 22, 24, 27, 28, 30, 32, 38

• Section 3.3, pages 211-212:
  • 4, 10, 12, 20, 21

• Maple Exercises:
  [See the posted The Subspaces of a Matrix and their Bases and Linear Transformations and Matrix Algebra.]

  Instructions for the Maple Exercises:

  • Staple these Maple Exercises to the end of all your handwritten work.
  • Seed the random number generator with your student ID number, as illustrated in Linear Transformations and Matrix Algebra. (Each student should be working with unique random matrices. If the TA notices two students have the same matrix, it will be assumed they copied from each other.)
  • Write explanations in your Maple work, describing what you are doing and why. Don't just hand-in a string of Maple computations with no explanation.
  • Clearly label each Maple problem.

  The Maple Exercises:

  • jh1: Starting from a random 5 by 5 matrix B, find the linear transformation which cyclically permutes the column vectors of B, and use this to construct a matrix A which is a 5th root of the 5 by 5 identity matrix. Evaluate A^5 to make sure you constructed it correctly.

  • jh2: Again starting from a random 5 by 5 matrix B, construct a matrix A which is a 4th root of the 5 by 5 identity matrix. (Use a similar method to that shown in Linear Transformations and Matrix Algebra for finding a 4th root of the 3 by 3 identity matrix.) Evaluate A^4 to make sure you constructed it correctly.

  • jh3: Once more starting from a random 5 by 5 matrix B, construct a matrix A which is a 5th root of the 5 by 5 zero matrix. (Use a similar method to that shown in Linear Transformations and Matrix Algebra for finding a 3rd root of the 3 by 3 zero matrix.) Evaluate A^5 to make sure you constructed it correctly.

  • jh4: The posted Maple worksheet Linear Transformations and Matrix Algebra showed how to find a general 3 by 3 matrix of rank 2 which is its own square. Can you find a genaral 3 by 3 matrix of rank 1 which is its own square? If so, construct it and check that it works.