## Mathematics 510: Linear Algebra

Instructor: Jonathan Smith, 496 Carver, 4-8172 (voice mail)

e-mail: jdhsmithATmathDOTiastateDOTedu (substitute punctuation)

Office Hours: Mondays 10am-12noon, 2-3pm, 5:20-6:20pm; Wednesdays 10-11am (subject to change).

Grading: based on three graded homework assignments (45%), mid-term [Monday March 10, 9-9:50am, Carver 4] (25%), and final [Monday May 5, 7:30-9:30am, Carver 4] (30%).

Textbook: R.A. Horn & C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-30586-1.

Study plan: reserve 1 - 2 hours for homework between each pair of classes. Successful performance in the class depends critically on completion of the homework assignments.

Syllabus:

1. Aspects of undergraduate linear algebra:
1. matrix arithmetic, reduced row ecehelon form of a matrix, general solution to a system of linear equations;
2. determinants and their properties;
3. vector spaces: subspaces, basis, coordinate vectors, change of basis;
4. linear transfomations: matrix of a transformation, kernel, range, rank;
5. inner products: orthonormality, Gram-Schmidt, projection.
These concepts are reviewed as they arise in connection with the main topics listed below.
2. Eigenvalues, eigenvectors, characteristic polynomial, minimal polynomial, Cayley-Hamilton Theorem, algebraic and geometric multiplicity, diagonalization.
3. Unitary matrices and transformations, normal matrices and transformations, unitary diagonalization of normal matrices, Spectral Theorem, Schur's Unitary Triangularization Theorem.
4. Hermitian matrices, Rayleigh-Ritz Theorem, variational characterization of eigenvalues (min-max) and applications.
5. Positive-definite matrices, polar decomposition, singular value decomposition.
6. Jordan canonical form.

### Assignments

4/23 for 4/25: 3.2 7, 8, 9.
4/21 for 4/23: 3.2 3, 6.
4/18 for 4/21: 3.1 1, 3.2 5.
4/16 for 4/18: Exercise towards bottom of page 91.
4/11 for 4/16: 7.3 1, 3.

Three questions from 4.1 6, 11, 4.3 15, 7.1 1, 4.

4/2 for 4/4: Three Exercises at bottom of page 404.
3/31 for 4/2: 4.3 5, 16.
3/28 for 3/31: 4.2 1.
3/26 for 3/28: 4.2 2, 3.
3/24 for 3/26: 4.1 1 ("Hermitian" part only), 4, 10.

3/3 for 3/5: 2.5 8.
Also: prove that G, H Hermitian (and of the same size) imply (1/2i)(GH-HG) Hermitian.
2/28 for 3/3: 2.5 1, 15, 25 ("complex" part only).
2/26 for 2/28: 1.3 11.
2/21 for 2/26: 1.4 1, 4, 10, 1.1 8, Exercise at top of page 60.

Three questions from 1.2 4, 8 (prove the "assumption" as well), 2.1 7, 12, 2.2 8.

2/12 for 2/14: 2.1 1, 2, 3, 4, 5.
2/10 for 2/12: Exercise at bottom of page 66.

Three questions from 1.3 6, 7, 5, 8, 10.

1/29 for 1/31: Exercise at bottom of page 45, then first three Exercises at top of page 46.

1/27 for 1/29:
1. Fix a positive integer n. Let Pn be the set of all polynomials
p(t) = antn + ... + a1t + a0
of degree at most n, with real coefficients an , ... , a0 .
1. Show that Pn forms a real vector space under addition and constant multiplication of polynomials.
2. Show that B = { t j |   j = 0, ... , n } and B' = { (t+1) j |   j = 0, ... , n } are bases of the vector space Pn.
3. Determine the change-of-basis matrices B[ I ]B' and B'[ I ]B .
2. Repeat Question 1(iii), replacing the field of real numbers by the two-element field Z2 .

1/22 for 1/27:
1. Show that a set {v1,...,vn} of (column) vectors in Fn is a basis iff the n x n matrix
B = [v1| ... |vn]
is non-singular.
2. Show that the set of all real 2 x 2 matrices of shape
 [ x -y ] [ y x ]
forms a field.
3. Show that the set of all rational 2 x 2 matrices of shape
 [ x 2y ] [ y x ]
forms a field.

1/15 for 1/22: 1.1 3, 4, 5. 1.2 3, 7.
1/13 for 1/15: 1.1 1, 2.