Math 311-200 — Test 1 Review

Extra office hours. I will have extra office hours on Thursday, from 10-12 and on Friday morning, from 10-11.

Material covered.

• Notes on Row Reduction
• The Rank of a Matrix
• The Inverse of a Matrix
• Determinants
• Chapter 1 and Chapter 2, sections 2.1 through 2.3 in the text. Here are extra problems from these sections. (These will be part of the next homework assignment.) Section 2.1 — 8, 9; section 2.2 — 5, 8, 12; section 2.3 — 6, 9, 10.

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra or storing programs or other material.

Notation. You should know the notation that we have have used for row reduction, various vector spaces, coordinate vectors, etc.

Structure. There will be 4 to 6 questions, some with multiple parts. The questions will be similar to ones done for homework, quizzes, or as examples in class or in the text. There will be one ``theory'' question taken from the following list.

1. Be able to use the permutation definition of determinant to solve a problem similar, but not necessarily identical, to the one in the homework.
2. Be able to prove this representation theorem: If a vector space V has a basis B = {v₁, ..., v_n}, then every v in V can be represented in one and only one way as v = c₁ v₁ + ... + c_n v_n, where the c_j's are scalars.
3. Be able to show that if a vector space V has a basis B = {v₁, ..., v_n}, then every set with n+1 or more vectors is linearly dependent.

Updated 2/22/06