# Math 261 – Linear Algebra, Fall 2012

Instructor: Anatolii Grinshpan
Office hours:  M 10-12, Korman 247, or by appointment.

Course information   Schedule   Academic Calendar       Some links: Matthews, Strang, Terrell, Tutoring center

Sep. 26. Introduction. Systems of linear equations. Elementary row operations.

Sep. 28. Matrix notation. Row reduction. Echelon form. Pivot entries. Basic and free variables. Gaussian elimination algorithm.

Reading: 1.1 & 1.2. Problems: 2, 4-8, 11, 12, 15-17, 21-24, 26, 29-34 (pages 10-11),   1-6, 8, 11, 14-17, 21, 22, 24, 25, 29, 31 (pages 21-23)

Oct. 3. Reduced echelon form. Algebra of vectors. Linear combinations.

Oct. 5. Span. Description of solutions. Matrix-vector product.

Reading: 1.3 & 1.4. Problems: 5-7, 10, 11, 14, 16, 20, 22-24 (pages 32-33),  1-8, 11, 14, 15, 23, 24, 29, 31, 32, 37 (pages 40-42)

Oct 10. Matrix-vector product. Homogeneous linear systems.

Oct 12. Nonhomogeneous linear systems. Examples.

Reading: 1.5 (1.6 optional). Problems: 1-8, 11, 13, 15, 17, 19, 23, 24, 28-31, 34, 36, 37, 40 (pages 47-48)

Oct 15. Questions session: 6-7 PM, CAT 61.

Oct. 17. Midterm 1:  8-8:50 AM, PISB 120 (lectures of Sept 26 - Oct 12). Lecture: linear independence.

Reading: 1.7. Problems: 1-4, 6, 9, 15-24, 26, 27, 30, 33-41.

Reading: 1.8, 1.9. Problems: 1, 3, 5, 9, 11, 13-17, 19-22, 30, 31, 33, 34, 36 (pages 68-70), 1-11, 15-20, 23-32 (pages 78-79)

Oct. 24. Operations with matrices. Matrix product.

Reading: 2.1. Problems: 1, 3, 5, 7, 9, 11, 15-22 (pages 100-101)

Oct. 26. Matrix product. The transpose of matrix. Invertible linear transformations. The inverse of a matrix. Two-by-two case. Example. Example.

Oct. 31. Properties of matrix inversion. Matrices representing elementary row operations and their inverses. Inversion algorithm. Notes on inversion.

Reading: 2.2, 2.3. Problems: 1-7, 9, 10, 13, 15, 16, 21-24, 30, 32-35, 37 (pages 109-110), 1-9, 11-14, 16-24, 33 (pages 115-116).

Nov. 2. Characterizations of invertibility. Multipliers of row reduction. LU factorization.

Nov. 7. LU factorization. Permutation matrices. Pivoting. LDU factorization. Notes on LU.

Reading: 2.5. Problems: 2, 5, 7-10, 15, 17, 19, 20, 25 (pages 129-130).

Nov. 9. Partitioned matrices. Column-row expansion.

Reading: 2.4. Problems: 1, 3, 5-8, 10, 12, 13, 15 (page 121).

Nov 12. Questions session: 6-7 PM, Disque 103.

Nov. 14. Midterm 2:  8-8:50 AM, PISB 120 (lectures of Oct 17 – Nov 9). Lecture: row, column, and null spaces of a matrix. Rank of a matrix.

Nov. 16. Rank-nullity theorem. Basis and coordinates. Dimension of a subspace. A basis for the null space.

Reading: 2.8, 2.9. Problems: 1-5, 7, 15, 17, 21-26, 31-36  (pages 151-152), 1-3, 5, 7, 9-24  (pages 157-159).

Reading: 3.1, 3.2. Problems: 1, 3, 5, 11, 14, 19-21, 25, 29 (pages 167-168), 7, 12, 13, 16-19, 22, 24, 27-29, 31-36 (pages 175-176).

Nov. 30. Eigenspaces. Diagonalization. Examples.

Reading: 5.1-5.3. Problems: 1-5, 9, 10, 13, 17-19, 21, 23, 24, 26, 27 (pp 271-72), 1, 3, 5, 9, 10, 17-22 (pp 279-80), 2, 3, 5, 7, 9, 15, 17, 20-24, 27, 28, 31, 32 (pp 286-87).

Dec. 5. Diagonalizable matrices. Length and angle in n dimensions. Orthogonality. Eigenvalues in matlab.

Dec. 7. Orthogonal matrices. Symmetric matrices.

Reading: 6.1, 6.2. Problems: 7, 10, 14, 16, 22,  24 (pp 336-37), 1, 2, 8, 9, 17, 18, 21, 27 (pp 344-45).

Dec 7. Questions session: 5-7 PM, Randell 121.

Supplementary practice questions: Set 1 Set 2 Set 3 Set 4 Set 5 Set 6

Dec. 10. Final exam: 10:30-12:30, PISB 120 (all lectures).