Category Archives: linear algebra

Exercise 1.4.2. Multiply the matrices below: Work by columns instead of by rows. Answer: The first example multiplies a 3×2 matrix by a 2×1 matrix (column vector), producing a 3×1 matrix (column vector). Working by rows gives us the following: … Continue reading →

Exercise 1.4.1. Multiply the matrices below: (The exercise also asks you to draw a graph showing addition of the two vectors (2, 1) and (0, 3); I’m skipping that part.) Answer: The first example multiplies a 3×3 matrix by a … Continue reading →

Exercise 1.3.13. We have two sets of people, those who start the year residing in California and those who do not. 80% of those starting the year in California are still in California at the end of the year, while … Continue reading →

Exercise 1.3.12. We have two sets of people, those who start the year residing in California and those who do not. 80% of those starting the year in California are still in California at the end of the year, while … Continue reading →

Exercise 1.3.11. Given the systems of equations    and    solve both systems using Gaussian elimination. Answer: We start with the first system of equations The first elimination step produces The second elimination step produces We then back-substitute, starting with … Continue reading →

Exercise 1.3.10 (very optional). Find a method for computing the quantities ac – bd and bc + ad with three multiplications instead of four. Assuming that addition were a sufficiently faster operation than multiplication, this would provide a faster way … Continue reading →

Exercise 1.3.9. State whether the following statements are true or false. (Note that without loss of generality we can assume that no row exchanges occur during the process of elimination.) (a) Given a system in u, v, etc., where the … Continue reading →

Exercise 1.3.8. Given a system of equations of order n = 600, how long would it take to solve in terms of the number of multiplication-subtractions? In seconds, on a PC capable of 8,000 operations per second? On a VAX … Continue reading →

Exercise 1.3.7. (a) Given a system of equations A, with the first two rows the same, at what point in elimination will it become clear that A is singular? Show a 3×3 example. (b) Repeat (a), but instead assume that … Continue reading →

Exercise 1.3.6. Given the following system of equations: for what values of would Gaussian elimination break down, either in a fixable way (i.e., via row exchange) or in a non-fixable way? Answer: One way for elimination to break down temporarily … Continue reading →