MATH 212B
AUIA Summer Program
MAT 21: Linear Algebra
SYLLABUS
Instructor: Aaron Silberstein
Home institution: Harvard University
Email: asilbers@math.harvard.edu
Office Hour: TBA
Teaching Assistant: TBA
Course Description:
This is a basic course in linear algebra, with emphases on conceptual
understanding, visual intuition, and calculation. We begin by defining vector
spaces and linear maps, and introduce matrices as a way of computing with
these objects. We then move on to the notions of length, angle, and area, and
develop the algebraic structures behind these, with an emphasis on calculation.
After this, we study the procedures of diagonalization and row/column reduction
for matrices, ending with the Jordan normal form. The last week will be spent
on special topics and presentations, to be determined in class. The course is
synchronized with the multivariable course and, while independent, the two
courses will complement each other. Problem solving is a large component of
the course, and one lecture each week will be devoted to problem sessions.
Required Textbooks:
Halmos, Finite Dimensional Vector Spaces.
Grading & Evaluation:
Problem Sets: 70%.
Presentations: 30%.
Course Schedule:
Week 1: Real numbers and complex numbers; finite-dimensional real and
complex vector spaces; bases and dimension; subspaces and quotient spaces;
linear transformations and matrices.
Week 2: Length, angles, and area; inner products; direct sums; dual spaces
and adjoints; Gram-Schmidt orthogonalization; eigenvectors and eigenvalues –
definition, examples in dimension 2.
Week 3: Similarity and Jordan normal form; Gaussian elimination and echelon
forms; multilinear algebra.
Week 4: Eigenvectors and eigenvalues revisited: traces, determinants and
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MATH 212B
characteristic polynomials; Cayley's theorem; the division algorithm; solving
polynomials; the spectral theorem.
Week 5: Symmetries. Special topics (to be decided in-class). Presentations.
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