MAT329
Review for Exam #1
The first exam will cover material from sections 1.1-1.5, 1.7-1.10, and 2.1-2.3. This
handout is to serve as a basic review. The best way to prepare for the exam is to carefully
read the text, rework all in-class worksheets, quizzes, and homework assignments. In
addition, be sure to know all definitions as well as statements of major theorems.
After studying for the exam you should be able to:
1. use elementary row operations to put a matrix in row echelon form and reduced
row echelon form.
2. determine when a system of linear equations has a unique solution, no solution, or
infinitely many solutions and find the solution if it exists.
3. determine if a system of linear equations is consistent or inconsistent.
4. determine values of h which makes a linear system consistent or which makes a
collection of vectors span, or which makes a collection of vectors linearly
dependent.
5. understand pivot positions, pivot columns, free variables, and basic variables
6. define appropriate vectors to restate a system of linear equations in terms of linear
combinations of vectors and restate in terms of a matrix equation or vice versa.
7. solve a vector equation and a matrix equation.
8. write a solution in parametric vector form.
9. determine if a vector is a linear combination of a set of vectors.
10. restate a system of linear equations in terms of Span {v1 , v2 ,..., vn } and then solve.
11. restate a system of linear equations in terms of the columns of a matrix to
determine if the linear system is consistent.
12. compute A x
13. know when Ax  b has a solution in terms of span and pivot positions.
14. determine and justify when a vector is in the span of a set of vectors.
15. determine and justify if a set of vectors spans R n for a small value of n .
16. know when a homogeneous system of equations has a solution.
17. give a geometric description of the solution set of a system of linear equations.
18. determine if a system of homogeneous equations has a nontrivial solution
19. give a parametric description of all solutions of Ax  b and Ax  0
20. determine if a set of vectors are linearly independent or dependent.
21. determine by inspection (no computations) if a set of vectors are linearly
independent or dependent.
22. find a linear dependence for a set of vectors.
23. find an image of a vector x under a linear transformation.
24. determine if a vector b is in the range of a linear transformation.
25. given a linear transformation, find the unique matrix corresponding to it.
26. determine the dimensions of a matrix for a given linear transformation.
27. find all vectors x that get mapped to the zero vector.
28. give the two properties that make a transformation a linear transformation.
29. when given a linear transformation, find a formula for the image of a vector x.
30. determine if a linear transformation is onto.
31. determine if a linear transformation is one-to-one.
32. do an application problem involving a transition matrix – like something on the
Maple 2 lab.
33. find the matrix associated with a linear transformation.
MAT329
Review for Exam #1
34. determine if statements involving matrix properties are true or false.
35. justify true statements using matrix properties.
36. know how to calculate the determinant of a 2x2 matrix.
37. compute the inverse of an 2  2 matrix by hand and use Maple to find the inverse
of nxn matrices..
38. use the inverse of a matrix to solve a system of equations.
39. justify true statements using matrix properties.
40. recognize and use the connections in the “Big Theorem” to justify statements.
41. be able to do proofs of the “follow-your-nose” type. By this I mean you should
be able using just definitions to prove simple statements.