MATH 262, Review for Midterm Test 1

Test topics

1. Vector Spaces
   1. Sets and Logic (1.1)
   2. Definition of Vector Space (1.2)
   3. Properties of Vector Spaces (1.3)
   4. Subtraction and Cancellation (1.4)
   5. Euclidean Spaces (1.5)
   6. Matrices (1.6)
   7. Function Spaces (1.7)
   8. Subspaces (1.8)

2. Systems of Linear Equations
   1. Notation and Terminology (2.1)
   2. Gaussian Elimination. Gauss-Jordan Method (2.2)
   3. Solving Linear Systems (2.3)

3. Dimension Theory
   1. Linear Combinations (3.1)

Key Concepts

• Logical operations, sets, and set operations. Set notation. Set elements and subsets and the corresponding notations.
• Axioms of Vector Space: Know which properties are axioms, and which are derived from them.
• Examples of Vector Spaces: Rⁿ, Matrices, Polynomials, Functions.
• Subspaces and Theorem 1.11. Examples of subspaces and non-subspaces.
• Gaussian Elimination (forward) and the Gauss-Jordan Method (forward and backward).
• REF and RREF.
• Solution of Linear Systems using Gauss or Gauss-Jordan. The cases of one solution, no solutions, and infinitely many solutions.
• Linear Combinations: when is a vector a linear combination of other vectors.

Typical Questions

Proofs of Properties of Vector Spaces from Theorems 1.5 and 1.7 may be asked as part of the test. Questions from the homework and practice problems may be incuded as well.

General rules

• All tests are closed books/notes; graphing calculators, cell phones or other electronic devices are not permitted.
• The duration of the test is 75 minutes.
• Problems will come in the same format as on the quizzes: you will need to present a complete solution of the problem showing all steps to receive full credit.