Math 304 - Spring 2000

Review Sheet - Final Exam

General Information

The final exam is comprehensive. It covers chapters 1-5 (except for sections 2.3 and 5.7), and it also covers sections 6.1 and 6.3 in chapter 6. The structure of the final exam will be similar to the in-class tests, except that it will have a few more questions.

The distribution of points on the exam will be roughly this. Chapters 1 & 2, 10-15 points. Chapter 3, 20-25 points. Chapter 4, 20-25 points. Chapter 5, 25-30 points. Chapter 6, 10-15 pts. Extra points are assigned to the last two chapters because they have not been covered in an in-class test.

Chapters 1-4

See the reviews for Tests I and II, the tests themselves, quizzes 1-7, and the homework for these chapters. Very elementary material (matrix multiplication, for example) will be tested on only in so far as it comes up in connection with other material.

Chapter 5

Section 5.1. Scalar product in Rⁿ; scalar and vector projections; length and angle. Skip pp. 202-206.
Section 5.2. Orthogonal subspaces; orthogonal complements; direct sum; fundamental subspaces for a matrix A in R^{m
× n}
1. R(A) - the range of A (column space)
2. N(A) - the null space of A
3. R(A^T) - the range of the transpose of A (row space)
4. N(A^T) - the null space of the transpose of A
Know how these are related. For example, R(A) and N(A^T) are orthogonal complements, and the direct sum of these spaces is Rⁿ.
Section 5.3. Inner products, inner product spaces, Cauchy Schwarz inequality, norms.
Section 5.4. Least squares problems, normal equations,
Section 5.5. Orthogonal and orthonormal sets of vectors; orthogonal matrices; connection with least squares. Skip pp. 247-253.
Section 5.6. Gram-Schmidt procedure. Skip pp. 261-265.

Chapter 6

Section 6.1. Eigenvalues of a matrix; eigenspaces; characteristic polynomial; sum and product of eigenvalues.
Section 6.3. Diagonalizing a matrix; defective matrices; applications to Markov chains. Skip pp. 311-316.