Homework 1 - Due: 01/25
Problem 1
Which of the following are vector spaces over
- The set of real valued
matrices with nonnegative entries where is a given positive integer. - The set of rational functions of the form
where and are polynomials in the complex variable and the degree of does not exceed a given fixed positive integer . - The space
of square-integrable functions, i.e., functions with the property that
Problem 2
Let
Problem 3
Let
- Prove that
(the sum of the dimension of the nullspace of and the dimension of the range of equals the dimension of ). - Now assume that
. It is not always true that is a direct sum of and . Find a counterexample demonstrating this. Also, describe a class of linear transformations (as general as you can think of) for which this statement is true.
Problem 4
Consider the standard RLC circuit, except now allow its characteristics
Problem 5
Three employees — let’s call them Alice, Bob, and Cheng — received their end-of-the-year bonuses which their boss calculated as a linear combination of three performance scores: leadership, communication, and work quality. The coefficients (weights) in this linear combination are the same for all three employees, but the boss doesn’t disclose them. Alice knows that she got the score of 4 for leadership, 4 for communication, and 5 for work quality. Bob’s scores for the same categories were 3, 5, and 4, and Cheng’s scores were 5, 3, and 3. The bonus amounts are $18000 for Alice, $16000 for Bob, and $14000 for Cheng. The employees are now curious to determine the unknown coefficients (weights).
- Set up this problem as solving a linear equation of the form
for the unknown vector . - Calculate the unknown weights. It’s up to you whether you use part (a) for this or do it another way.
- Are the weights that you computed unique? Explain why or why not.