Review List For Exam #1 (October 8, 2007)

The Exam will cover the review material from chapters 1 and 2 and Sections 10.1-10.5.

Definitions:

Echelon form, Pivot positions.

Span of a set of vectors, Linear dependence and independence.

Linear transformations, Matrix of a linear transformation.

Theorem 1 (uniqueness of reduced echelon form).

Theorems 4,6,7,8,10,12.

p.48, Ex.17-20; p.55, Ex.7; p.102, Ex.1,7.

Chapter 2

Definitions:

Matrix multiplication, Inverse of a matrix, Partitioned matrices.

Subspace of Rⁿ.

Column space, Row space and Nullspace of a matrix.

Rank of a matrix.

Theorem 6.

Theorem 8, The Invertible Matrix Theorem and its additions in section 2.9.

Theorem 14, The Rank Theorem.

p.139, Ex.8; p.181, Ex.9; p.183, Ex.1.

Chapter 10

Probability vector, Stochastic matrix, Regular stochastic matrix, Steady state vector.

Communication classes, Reducible and irreducible Markov chains.

Transient classes and.states, Recurrent classes and states.

Canonical form of a transition matrix.

The fundamental matrix for a Markov chain whose transition matrix is in canonical form.

Theorems 1, 2, and 3 (Properties of regular transition matrices and irreducible Markov chains.)

Theorem 6 (Information provided by the fundamental matrix.)

p.296, Ex. 1; 10.1, Ex. 7; 10.2, Ex. 5; 10.3, Ex. 4; 10.4, ex. 4; 10.5, Ex. 1,7.