Homework

Homework is due Fridays by 3pm at your GSI's office or mailbox. Please follow your individual GSI's instructions as to where to turn it in.

You are encouraged to discuss ideas with other students. However, you must write and hand in your solutions independently.

Each week, two selected problems from the homework assignment will be graded. Solutions to all problems will be posted.

When calculating grades, we will drop your two lowest homework scores and use only your remaining scores.

Syllabus

1. Vector spaces.
2. Subspaces. Intersection, sum. Direct sum.
3. Span, linear independence, and bases.
4. Dimension of a finite-dimensional vector space.
5. Linear maps.
6. Nullspace, range and rank of a linear map.
7. Matrix of a linear map.
8. Invertible linear maps.
9. Eigenvalues and eigenvectors.
10. Inner product spaces.
11. Orthonormal bases and the Gram-Schmidt procedure.
12. Orthogonal projections; applications.
13. Adjoints.
14. Self-adjoint and normal operators; spectral theorem.
15. Operators on complex vector spaces.
16. Characteristic polynomials and minimal polynomial.
17. Jordan form.

Lecture 1 (8/23/12): Fields, Vector Spaces

Key concepts: field axioms; basic properties of fields; examples: rational, real, and complex numbers; examples and non-examples of finite fields; characteristic of a field; vector space axioms.

Reading: Section 1.2, Appendix C.

Lecture 2 (8/28/12): Vector Spaces, Subspaces

Key concepts: row vectors, column vectors, matrices; functions, polynomials; subspace characterizations; intersection of subspaces; transpose of matrix, symmetric and skew- or anti-symmetric matrices; trace of matrices, diagonal matrices; upper/lower triangular matrices.

Reading: Section 1.2, 1.3.

Lecture 3 (8/30/12): Subspaces, Linear combinations, Spans.

Key concepts: sum of subspaces; direct sum of vector spaces; linear combinations; span of a subset; generating sets.

Reading: Section 1.4, 1.5.

Lecture 4 (9/4/12): Linear dependence and independence, Bases.

Key concepts: linear dependence and independence; bases.

Reading: Section 1.5, 1.6.

Lecture 5 (9/6/12): Finite bases and dimension.

Key concepts: finite bases; constructing generating sets; finding bases; Replacement Theorem; dimension.

Reading: Section 1.6.

Lecture 6 (9/11/12): Linear transformations.

Key concepts: maps of sets; linear maps of vector spaces; kernels and images; nullity and rank, Dimension Theorem.

Reading: Section 2.1.

Lecture 7 (9/13/12): Properties of linear transformations, Matrices.

Key concepts: injective, surjective, bijective maps of sets; isomorphisms of vector spaces; coordinates with respect to a basis; matrices with respect to bases; the vector space of linear transformations.

Reading: Section 2.1, 2.2.

Lecture 8 (9/20/12): Composition of linear transformations, Inverses.

Key concepts: compositions of maps; basic properties of compositions; multiplication of matrices; isomorphisms and inverses; every finite dimensional vector space is isomorphic to coordinate space.

Reading: Section 2.3, 2.4.

Lecture 9 (9/25/12): Change of bases, Dual spaces.

Key concepts: change of coordinate matrices; similar matrices; dual spaces and dual bases.

Reading: Section 2.5, 2.6.

Lecture 10 (9/27/12): Elementary matrix operations.

Key concepts: elementary matrices and operations; rank of matrix; simplifying matrices; calculating inverses.

Reading: Section 3.1, 3.2.

Lecture 11 (10/2/12): Solving systems of linear equations.

Key concepts: consistent and inconsistent systems; homogenous and nonhomogenous systems; relation of solutions and null space; criteria for existence of solutions; Gaussian elimination and reduced row echelon form; solving systems of linear equations in reduced row echelon form.

Reading: Section 3.3, 3.4.

Lecture 12 (10/4/12): Demystifying dual spaces.

Key concepts: dual spaces; dual bases; transposes.

Reading: Section 2.6.

Lecture 13 (10/9/12): Determinants.

Key concepts: geometry of determinants of 2x2-matrices; inductive definition of determinants in general; linearity with respect to fixed row; effect of row operations.

Reading: Section 4.1, 4.2.

Lecture 14 (10/11/12): Determinants of products.

Key concepts: determinant of product of matrices is product of determinants; determinant is nonzero if and only if matrix is invertible; determinants of elementary matrices; determinant of transpose; effect of column operations.

Reading: Section 4.3.

Lecture 15 (10/18/12): Eigenvalues and eigenvectors.

Key concepts: eigenvectors and eigenvalues; diagonalizability; characteristic polynomial; eigenvalues equal roots of characteristic polynomial.

Reading: Section 5.1.

Lecture 16 (10/23/12): Diagonalizability.

Key concepts: algorithm to check diagonalizability; split or non-split polynomials; multiplicity of roots of characteristic polynomial; dimension of eigenspace.

Reading: Section 5.2.

Lecture 17 (10/25/12): More on diagonalizability.

Key concepts: examples of all possible outcomes of algorithm; comparison of multiplicity of eigenvalue to dimension of eigenspace; applications of diagonalization.

Reading: Section 5.2.

Lecture 18 (11/1/12): Cayley-Hamilton Theorem.

Key concepts: invariant subspace; generating vector; characteristic polynomial of block upper triangular matrix; Cayley-Hamilton Theorem.

Reading: Section 5.4.

Lecture 19 (11/6/12): Complex numbers and inner products.

Key concepts: review of complex numbers; Fundamental Theorem of Algebra; definition of inner product; inner products from matrices.

Reading: Section 6.1, Appendix D.

Lecture 20 (11/8/12): Properties of inner products.

Key concepts: basic inequalities; orthogonal sets; unit vectors; orthonormal sets.

Reading: Section 6.1.

Lecture 21 (11/13/12): Gram-Schmidt process.

Key concepts: Gram-Schmidt process; orthonormal bases; coefficients with respect to orthonormal bases; orthogonal complements; direct sum of subspace and its orthogonal complement.

Reading: Section 6.2.

Lecture 22 (11/15/12): Adjoints.

Key concepts: orthogonal projections; existence and uniqueness of adjoint of linear transformation; compatibility with conjugate-transpose of a matrix.

Reading: Section 6.3

Lecture 23 (11/20/12): Self-adjoint transformations.

Key concepts: relation of eigenvalues for transformation and its adjoint; Schur's theorem; Spectral theorem; inner products are self-adjoint matrices with positive eigenvalues; orthogonal and unitary transformations.

Reading: Section 6.4, 6.5, 6.6.

Lecture 24 (11/27/12): Jordan canonical form I.

Key concepts: definition of Jordan canonical form; notions of generalized eigenvector and eigenspace; dot pictures organizing Jordan canonical form data.

Reading: Section 7.1, 7.2.

Lecture 25 (11/29/12): Jordan canonical form II.

Key concepts: decomposition into generalized eigenspaces; cycles of generalized eigenvectors; minimal polynomial.

Reading: Section 7.1, 7.2, 7.3

Useful Resources

Some previous Math 110 course home pages: