# Math 54 (Spring '09)

## Sections 109 and 112

**Instructor:** Prof. Paul Vojta

**Course website:** http://math.berkeley.edu/~vojta/54.html

**Homework assignments:** http://math.berkeley.edu/~vojta/syll.html

**GSI:** James Tener

**Section Information Sheet:** PDF

**Office:** 848 Evans

**Office hours:** Tuesday 3:15-4:15, Thursday 1:30-2:30, and by appointment.

**Meeting times:** - Section 109: MWF 11-12 in 70 Evans
- Section 112: MWF 2-3 in 30 Wheeler

**Lesson summaries**

January 21: Administrative stuff, relationship between matrices and systems of equations, basic vs. free variables, inconsistent systems, existence and uniqueness theorem

January 23: Linear combinations, spans, Worksheet 1 (solutions)
January 26: Solution sets of nonhomogenous systems, linear independence: definition, characterization in terms of matrices, examples, tricks
January 28: Worksheet 2 (solutions), linear transformations: example, prove they are specified by their values on a spanning set.
January 30: Linear transformations - representation by matrices, one-to-one linear transformations. Quiz 1 - Solutions 109 | 112
February 2: Matrix multiplication: when is it defined?, Inverse matrices - determining if matrices are invertible, finding inverses, examples
February 4: Review HW problems, intro to subspaces
February 6: Invertible matrices in terms of: pivots, rank, null space, columns. Coordinates, Worksheet 3 (solutions)
February 9: Bases/subspaces/dimensions, continued.
February 11: Coordinates, tools for finding determinants (cofactor expansion, row/column operations, triangular matrices), Cramer's rule (112)
February 13: Determinant warmup, change in area under a linear transformation, Quiz 2 - Solutions 109 | 112
February 16: *President's Day*
February 17: Midterm review session - Review sheet, exercises, and solutions
February 18: Vector spaces and more vector spaces
February 20: Go over exams
February 23: More on vector spaces and linear transformations, Worksheet 4 (solutions)
February 25: Coordinate systems, intro to change of basis
February 27: Coordinate systems, change of basis
March 2: Change of basis continued, Worksheet 5 (solutions)
March 4: Review Chapter 4 (and HW 6)
March 6: Intro to eigenvalues/eigenvectors, Quiz 3 - Solutions 109 | 112
March 9: Characteristic equation, diagonalizability
March 11: Matrices for linear transformations, similarity of such matrices
March 13: Dot product, orthogonality, orthogonal and orthonormal bases, reconstruction formula
March 16: Orthogonal matrices, projections onto lines
March 18: Orthogonal complement (section 109), Quiz 4 - Solutions 109 | 112
March 20: Least-squares approximation, intro to inner products, Worksheet 6 (solutions)
March 23: *Spring break*
March 25: *Spring break*
March 27: *Spring break*
March 30: Inner product spaces continued, applications of the Cauchy-Schwarz inequality, review for midterm
April 1: Review for midterm, Review sheet, Problems, Solutions
April 3: Go over exam
April 6: Intro to differential equations, especially second order, linear, constant coefficient homogenous ODEs
April 8: Homogenous equations continued, and intro to method of undetermined coefficients
April 10: Method of undetermined coefficients continued
April 13: Intro to higher order linear ODEs, existence of solutions to initial value problems (6.1)
April 15: Review homework, higher order linear, homogenous constant coefficient ODEs (6.2)
April 17: Finish chapter 6, Quiz 5 - Solutions 109 | 112
April 20: Intro to systems of first order equations, their normal form, and representing higher order linear equations as systems of first order equations
April 22: Review homework, systems of homogenous constant coefficient, first order linear equations (Section 9.5)
April 24: Systems with complex eigenvalues, Quiz 6 - Solutions 109 | 112
April 27: Intro to partial differential equations, solving the heat equation
April 29: Review homework 13
May 1: Review homework, convergence of Fourier series
May 4: Convergence of Fourier series, even and odd reflection, Fourier sine and cosine series.
May 6: Solving heat problems with Fourier sine series, heat problems with insulated endpoints
May 8: Heat problems with non-homogenous boundary conditions, Quiz 7 - Solutions 109 | 112
May 11: Discuss final, finding formal solutions to simple wave equations, differential equations review sheet