Monthly Archives: May 2010

Exercise 1.3.1. Solve the following equation using Gaussian elimination: Answer: The first pivot is 2 (the coefficient of u in the first equation). We multiply the first equation by 2 (the coefficient of u in the second equation divided by … Continue reading →

Exercise 1.2.13. Given the equation x + 4y = 7 for a line in the x-y plane, find the equation for a line that is parallel to the first line and passes through the point (0, 0). The first line … Continue reading →

Exercise 1.2.12. We have two equations, x + y + z = 1 and x + y + z = 2. The first part of the exercise is to sketch the planes in 3-space associated with these two equations; I’m … Continue reading →

Exercise 1.2.11. Assume that we have the following system of two equations in two unknowns: Under what circumstances would this system have a set of solutions constituting an entire line in the x-y plane? Answer: Each of the equations corresponds … Continue reading →

Exercise 1.2.10. Assume we have the points , , and . What values must , , and have in order for these points to fall on a straight line? Answer: If the points fall on the same line then we … Continue reading →

Exercise 1.2.9. We can re-express the system of equations from 1.2.8 as follows: Demonstrate that the third column can be expressed as a linear combination of the first two columns, so that the three columns lie in the same plane. … Continue reading →

Exercise 1.2.8. We have the following system of three of equations in three unknowns: Show that the system is singular by showing that there is a combination of the three equations that produces the contradiction 0 = 1. Find a … Continue reading →

Exercise 1.2.7. Start with equation (4) from p. 8 of the text: where a solution can be found for b = (2, 5, 7) and cannot be found for b = (2, 5, 6). Find two more values of b … Continue reading →

Exercise 1.2.6. Start with equation (4) from p. 8 of the text: and assume that b = (2, 5, 7) (a value of b for which solutions can be found for u, v, and w). The text mentions two solutions, … Continue reading →

Exercise 1.2.5. The following three equations: correspond to planes in 4-space (or space-time). Find two points on the line that is the intersection of the planes represented by the equations. Answer: We substitute the values of t and z from … Continue reading →