Note that this is a study guide, not a sample exam - it is much longer than your exam will be. However, the ideas and the question types represented here (along with your homework) will help prepare you for your exam. This exam covers material from 1.1 to 1.9 (except 1.6). This study guide was originally prepared by Jeanette Martin, and modified for our class by me. Contact me if you have any questions. To recieve full credit for correct answers, it is necessary to
show
your work and/or provide your reasoning. When row operations are
needed,
indicate at each step which row operations were performed. No
programmable
or graphing calculators are allowed. Answers to these questions are here. 1. Consider the following systems of equations. For each system, (i) Write the system as a matrix equation.
2. Discuss whether the appropriate space (ie, R
3. Determine all
4. For what values of v, _{1}v}?
For what values of _{2}h is {v,_{1}v,_{2}v}
linearly dependent? _{3}
5. Determine the values of
6. Suppose (a) How many pivots must 7. Suppose (a) How many pivots must 8. Show that if v, _{2}v,
_{3}v,
and _{4}v are in R_{5}^{5 }and v=
_{3 }0,
then {v, _{1}v, _{2}v,
_{3}v, _{4}v}
is linearly dependent. _{5}9. Describe all solutions of
10. Determine whether each of the following systems is
consistent or
inconsistent. When possible, solve the system, and describe the
solution
set in parametric vector form. State whether the solution set is unique
or not unique.
- 12. Determine if the following matrices are in reduced
echelon
form, echelon form, or neither.
13. Reduce the matrix to reduced echelon form. 14. Find the general solutions of the system whose augmented matrix is given below. 15. Determine if
16. Let , , Is 17. Show that 18. Compute the product A
19. List five vectors in the Span {v1, v2}, showing the
weights on
20. Find a vector
21. How many rows and columns must a matrix A have in order to define a mapping from: (a) R (b) R 22. Find all A = 23. Let
Let T: R^{3} -> R^{3} be a linear
transformation
that maps x into x_{1}v + x_{2}w
+ x_{3}z.
(a) Find a matrix A such that T(x) = Ax
for each x. (b) Find the image of u =
under the transformation T. 24. Let
ind the images of
and
under 25. Let 26. Let 27. Let e_{1}
unchanged. Find the standard matrix of T. 28. Let (a) Does T map R 29. Let (a) Does T map R Look at the answers |