**Course** Description: | Theory and applications of systems of linear equations, vector spaces, and linear transformations. Fundamental concepts include: linear independence, basis, and dimension; orthogonality, projections, and least squares solutions of inconsistent systems; eigenvalues, eigenvectors, and applications to Markov chains, difference equations, and quadratic forms. |

**Course Objectives:** | This is a first course in linear algebra; it prepares students for future studies in
all areas of mathematics and for studies in applied sciences. Students should
understand the concepts of vector spaces, bases, linear transformation, and matrix
algebras. They should be able to deal with these concepts at both the abstract and
the concrete level.
Students should be able to do the following:
(1) Solve systems of linear equations by means of Gaussian elimination
(2) Calculate the inverse of a matrix
(3) Determine the projection operator onto a subspace
(4) Determine the null space and range space of a linear transformation
(5) Prove elementary facts about vector spaces and linear transformations |

**Topical Outline:** | 1. Vector algebra and geometry. Dot products.
2. Matrices, echelon forms, and solution of systems of linear
equations. Applications.
3. Matrix algebra: product, inverse, and transpose.
4. Subspaces: nullspace, rowspace, column space of a matrix.
5. Linear independence, basis, and dimension, orthogonal complement.
6. Linear transformations, projections and the least-squares
solution. Gram-Schmidt process. Change of basis formula.
7. Determinants: signed area and volume, relation to singularity.
8. Eigenvalues and eigenvectors, diagonalizability, applications
to Markov processes and differential equations. Spectral theorem. |