Format and topics for exam 3
General information. Exam 2 will be a timed test of 50 minutes, covering sections 3.1-3.2, 4.1-4.3, and 5.1-5.2 of the text. No books, notes, calculators, etc., are allowed. Most of the exam will rely on understanding the problem sets (including problems to be done but not to be turned in), the quizzes, and the definitions and theorems that lie behind them. If you can do all of the homework and the quizzes, and you know and understand all of the definitions and the statements of all of the theorems we've studied, you should be in good shape.
You should not spend time memorizing proofs of theorems from the book, but you should defintely spend time memorizing the statements of the important theorems in the text, especially any named theorems.
Types of questions. All four of the previously described types of questions (computations, statements of definitions and theorems, paragraph homework-style questions, and true/false with justification) will probably appear on exam 3.
Definitions. The most important definitions and symbols we have covered are:
3.1 determinant cofactor expansion on row i lower triangular upper triangular 4.1 subspace closed under vector addition closed under scalar multiplication zero subspace nonzero subspace null space Null A column space Col A row space Row A 4.2 basis standard basis of Rn dimension 5.1 eigenvector (linear operator) eigenvalue (linear operator) eigenvector (matrix) eigenvalue (matrix) eigenspace corresponding to eigenvalue lambda lambda-eigenvector lambda-eigenspace 5.2 characteristic equation (matrix) characteristic polynomial (matrix) characteristic equation (linear operator) characteristic polynomial (linear operator) multiplicity of an eigenvalue similar matrices
Examples. Make sure you understand the following examples.
Theorems, results, algorithms. The most important theorems, results, and algorithms we have covered are listed below. You should understand all of these results, and you should be able to cite them as needed. You should also be prepared to recite named theorems.
Types of computations. You should also know how to do the following general types of computations. (Note also that on the actual exam, there will be problems that are not of these types. Nevertheless, it will be helpful to know how to do all these types.)
Not on exam. 3.1: Geometric applications of the determinant. Also, you should know the definition of determinant for practical purposes (i.e., expansion on rows), but you will not have to recite the definition. 3.2: Cramer's rule.