Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
A linear algebra exam from february 4, 2004. It includes various problems related to matrices, echelon form, reduced echelon form, linear independence, and linear transformations. Students are required to show all their steps and calculations to earn partial credit.
Typology: Exams
1 / 5
This page cannot be seen from the preview
Don't miss anything!
Name:
While the final answer is important, you earn points for all the work leading to that answer, as well as the answer itself. Show all your steps clearly so you will be eligible for the most partial credit. Good luck!
1.) Provide examples of each of the following.
a.) (5 pts.) A 4 × 3 matrix, not in echelon b.) (5 pts.) A 3 × 4 matrix in echelon form, form but not reduced echelon form
c.) (5 pts.) A 4 × 4 matrix with exactly d.) (5 pts.) I 4 three pivots
2.) a.) (10 pts.) Compute the reduced echelon form of the matrix A =
SHOW ALL YOUR STEPS. You may use a calculator to check your work, but a calculator answer alone earns no credit.
b.) (5 pts.) Suppose A is an augmented matrix associated with a linear system. Write out the associated system of linear equations.
c.) (5 pts.) Is the system you wrote out in part (b.) consistent? Why or why not?
4.) (10 pts.) Let A be a 3×2 matrix. Explain why the equation Ax = b cannot be consistent for all b in R^3.
5.) (10 pts.) Show that the transformation T defined by T (x 1 , x 2 ) = (2x 1 − 3 x 2 , x 1 + 4, 5 x 2 ) is not linear.
6.) a.) (5 pts.) True or False: If A and B are 2 × 2 with columns a 1 , a 2 , and b 1 , b 2 , respectively, then AB =
a 1 b 1 a 2 b 2
. If true, explain why. If false, correct the statement to make it true.
b.) (10 pts.) Compute AB if A =
(^) and B =
. NOTE: many
of you can easily do this in your head, which is fine. To show work on this problem, write out all the details for ONE ENTRY of the matrix AB; it’s not necessary to write out details for every entry of AB.
