Monthly Archives: September 2011

Exercise 2.3.3. Given the general triangular matrix show that the rows of are linearly dependent if any of the diagonal entries , , or is zero. Answer: We can test for linear independence of the rows by doing elimination on … Continue reading →

Exercise 2.3.2. State whether the following sets of vectors are linearly independent or not (a) , , and (b) given any four vectors , , , and , the vectors , , , and (c) for any , , and … Continue reading →

Exercise 2.3.1. State whether the following vectors are linearly independent or not by solving the equation . Also, solve to determine whether the vectors span . Answer: The equation is equivalent to We can use elimination to solve this system … Continue reading →

Exercise 2.2.14. Create a 2 by 2 system of equations that has many homogeneous solutions but no particular solution. Answer: A trivial example of such a system is where and ; for example The corresponding homogeneous system has any vector … Continue reading →

Exercise 2.2.13. What is a 3 by 3 system of equations that has the following general solution (the same as in exercise 2.2.12) and that has no solution if ? Answer: As in exercise 2.2.12, the general solution above is … Continue reading →

Exercise 2.2.12. What is a 2 by 3 system of equations that has the following general solution? Answer: The general solution above is the sum of a particular solution and a homogeneous solution, where and Since is the only variable … Continue reading →

Exercise 2.2.11. Let be a system of equations in unknowns, and suppose that the only solution to this system is . In this case what is the rank of ? Answer: If the system has no solutions then the system … Continue reading →