Math 2135 Fall 2025

MATH 2135: Linear Algebra for Math Majors (Fall 2025)

Syllabus

Final exam

Wednesday, Dec 10, 7:30-10:00 pm, in class
topics
practice final [solutions]
final office hour: Tuesday, Dec 9, 1 - 2 pm, Math 310
discussion of results: Thursday, Dec 11, 2- 3 pm, Math 310

Schedule

Numbers are for orientation and refer to sections in Lay et al, Linear algebra and its applications.

08/22: some applications of linear algebra
08/25: systems of linear equations, matrix representation (1.1), matrix times vector
08/27: elementary row operations, row reduction, (reduced) echelon form (1.2)
08/29: free variables, solution in parametrized form, homogenous and inhomogenous systems
09/03: existence and number of solutions of Ax=b (1.2), vectors, linear combinations, span (1.3)
09/05: Ax=b is consistent iff b is in the span of the columns of A
09/08: solutions of homogenous and inhomogenous systems (1.5), nullspace of A
09/10: linear independent vectors (1.7)
09/12: linear transformations (1.8)
09/15: standard matrix of a linear transformation, every linear map is of the form x -> Ax (1.9)
09/17: characterizing injective, surjective x -> Ax by the columns of A (1.9)
09/19: matrix addition, composition of linear maps, matrix multiplication (2.1)
09/22: properties of matrix operations (2.1)
09/24: inverse matrix (2.2)
09/26: computing inverse matrix by row reduction (2.2), REVIEW [pdf], [practice midterm], [solutions]
09/29: MIDTERM 1
10/01: computations in Mathematica [available from CU to download and install] [Mathematica notebook], properties of inverse matrices (2.2)
10/03: characterizing invertible matrices (2.3) [pdf] [video]
10/06 axioms of vectors spaces and examples: tuples, matrices, sequences, functions as vector spaces (4.1), subspaces (2.8, 4.1)
10/08: discussion of midterm 1, spans and null spaces are subspaces (2.8, 4.2)
10/10: basis of subspaces, basis of null space (2.8, 4.3)
10/13: linear maps, linear independent sets, column space (4.3)
10/15: basis of column space, Spanning Set Theorem to remove vectors from a spanning set to obtain basis (2.8, 4.3)
10/17: basis of row space (4.6), coordinates relative to a basis (existence and uniqueness) (2.9, 4.4)
10/20: dimension of vector spaces (4.5), isomorphisms
10/22: every linear independent set extends to a basis (4.5)
10/24 rank of matrix (4.6), matrix of a linear map w.r.t bases B,C (cf 4.7)
10/27: reflection on line, integration matrix for polynomials
10/29: determinants, cofactor expansion by a row or column (3.1)
10/31: REVIEW [pdf] [practice midterm], [solutions]
11/03: MIDTERM 2
11/05: determinant by row reduction, A is invertible iff det A <> 0, det AB = det A det B (3.2)
11/07: determinant as area of parallelogram (3.3),
11/10: eigenvalues, eigenvectors, eigenspaces (5.1), characteristic polynomial of a matrix (5.2)
11/12: diagonalizable matrices, Diagonalization Theorem (5.3)
11/14: basis of eigenvectors (5.3)
11/17: dynamical systems (5.6), Fibonacci sequence
11/19: dot product, length of vectors over the reals, orthogonal vectors (6.1)
11/21: orthogonal complement of a subspace (6.1), orthogonal and orthonormal sets, orthogonal basis, coordinates via dot product (6.2)
12/01: orthogonal projection onto a vector, onto a subspace of R^n (6.3)
12/03: least squares solution of inconsistent systems (6.5)
12/05: REVIEW [practice final]

Homework

Handouts

Functions [pdf]