Review for Second Exam

MATH 240

The exam will cover the material we have discussed in class and studied in homework, from Chapter 3 and Sections 4.1 to 4.7. The following list points out the most important definitions and theorems. Note that when a computation is required, you may be asked to justify the conclusion you draw from the computation (by mentioning a fact or theorem, for example)

Definitions:

Vector space, subspace, null space, column space
Linear transformation
Linear independence, linear dependence, linear dependence relation (for vector spaces)
Spanning set, basis, coordinate vector
Dimension of a vector space
Rank of a matrix
Change of coordinates matrix

Theorems:

Chapter 3:

Theorems 2, 3, 4, 5, 6 (properties of determinants)
The formula for det A as a product of pivots (page 188)
Theorem 7 (Cramer's Rule)
Theorems 9 and 10 (determinants as area or volume)

Chapter 4:

Theorems 1-4, 5 (Spanning Set Theorem), 6, 7 (Unique Representation Theorem), 8-10, 12 (Basis Theorem), 13, 14 (Rank Theorem) and 15.
Know the proofs of Theorems 2 and 7.
The Invertible Matrix Theorem (including new statements in Sections 3.2 (Theorem 4) and 4.6)

Important Skills (partial list):

Compute a determinant using a combination of elementary row (or column) operations and cofactor expansions (across a row or down a column).
Compute a 3 by 3 determinant.
Determine if a set of vectors spans (or is a basis for) Rⁿ.
Determine if a set is a subspace (using Theorems 1, 2, or 3 in Chapter 4).
Determine if a vector is in Nul A or in Col A.
Find a nonzero vector in Nul A or Col A.
Determine if a set is a basis for a subspace.
Find the dimension of Nul A, Col A, Row A, or other subspace.
Find a basis for Nul A, Col A, Row A, or other subspaces.
Find the coordinate vector of a vector relative to a basis.
Find the change-of-coordinates matrix.
Use coordinate vectors to check if a set is linearly independent.
Determine the rank of a matrix.
Use the Rank Theorem to determine facts about a system of linear equations.

Applications:

Compute areas or volumes of simple objects using determinants.

Further Review:

After you have completed your initial review, try taking part of the sample Second Exam for Course A (only questions 1, 2, 5, and 6) on the Web at www.laylinalgebra.com (or on the CD). Take the exam in a quiet place, at a time when you can work without stopping, and without looking at the text or your notes. After that, check the solutions that accompany the sample exam. But, please don't look at the solutions until you have tried the entire test.

Remember, the exam may include material that is not on the sample exam, and the wording of questions may be somewhat different.

The Web/CD also contains the following additional exam questions that fit our coverage this semester:

Course B, Second Exam: Questions 3, 5, 6.
Course C, Second Exam: Questions 1, 3, 4, 5, 6.
All of the exams on the Web/CD are from exams given at the University of Maryland.