### Math 223 - Linear Algebra

Syllabus

We will cover all subsections in the textbook except the ones marked with asterisk (1.7, 2.6, 2.7, ...).
Inner product spaces (Section 6) will be covered if we finish Diagonalization early.
The timeline is as follows:

1. Vector Spaces.
2. Linear Transformations.

First Midterm Oct. 9.

3. Matrix operations.
4. Determinants.

Second Midterm Nov 13.

5. Diagonalization.

Final exam.

Homeworks:
Problem set #1, Solutions
Problem set #2, Solutions
Problem set #3, Solutions
Problem set #4, Solutions
Problem set #5, Solutions
Problem set #6, Solutions
Problem set #7, Solutions
Problem set #8, Solutions
Problem set #9, Solutions
Problem set #10, Solutions

First Midterm, Solutions

Topics covered:
• The concept of a vector space. The 8 axioms will not be needed, but you have to know the basic  examples of vector spaces  and the two operations in them.
• Subspaces, checking the three defining properties of a subspace.
• Span and linear independence, basis.
• Linear transformations
• Nullspace, range, the dimension formula.
Previous first midterms:

Second Midterm, Solutions.

Topics covered:
• Composition of linear transformations and matrix multiplication.
• Isomorphisms and invertible matrices.
• The matrix of a linear transformation.
• Gaussian elimination and everything we can find with it: solutions, inverse matrices, bases for the nullspace and range.
• Determinants. Computation of determinants using expansion or Gaussian elimination. Properties of determinants.
Previous second midterms:

Final Exam.

Topics covered:
• Everything from the midterms.
• Diagonalization of matrices: finding eigenvalues, eigenvectors, expressing A=PDP^{-1}
• Applications of diagonalization: Powers of matrices, population models, Markov chains, systems of linear differential equations (the main part here is computing the exponential of a matrix).
• Orthogonal vectors, Gram-Schmidt algorithm.
• Othonormal eigenbases for symmetric matrices (quadratic forms will not be included).
Previous final exams: