Linear Algebra

URL study guide

https://studiegids.vu.nl/en/courses/2024-2025/X_400649

Course Objective

All that's below falls under:
- Knowledge and understanding
- Applying knowledge and understanding
- Making judgements
- Communication
- Lifelong learning skills (since Linear Algebra appears in all of science and industry) Besides being able to explain, interrelate, know the basic properties of, and construct simple arguments with the concepts listed above, the student will learn the following skills (organized by topic): Linear systems:
- Can solve systems of linear equations using row-reduction
- Can determine the number of solutions of a linear system
- Can prove or disprove simple statements concerning linear systems Linear transformations:
- Can determine if a linear transformation is one-to-one and onto
- Can compute the standard matrix of a linear transformation
- Can use row-reduction to compute the inverse of a matrix
- Can prove or disprove simple statements concerning linear transformations Subspaces and bases:
- Can compute bases for the row and column space of a matrix
- Can compute the dimension and determine the basis of a subspace
- Can prove or disprove simple statements concerning linear systems Eigenvalues and eigenvectors:
- Can compute the eigenvalues of a matrix using the characteristic equation
- Can compute bases for the eigenspaces of a matrix
- Can diagonalize a matrix
- Can prove or disprove simple statements concerning eigenvalues and eigenvectors Orthogonality:
- Can compute the orthogonal projection onto a subspace
- Can determine an orthonormal basis for a subspace using the Gramm-Schmidt algorithm
- Can solve least-squares problems using an orthogonal projection
- Can orthogonally diagonalize a symmetric matrix
- Can compute a singular value decomposition of a matrix
- Can prove or disprove simple statements concerning orthogonality

Course Content

The topics that will be treated are listed below. For every topic, the relevant concepts are listed. Linear systems: linear system (consistent/inconsistent/homogeneous/inhomogeneous), (augmented) coefficient matrix, row equivalence, pivot position/column, (reduced) echelon form, basic/free variable, spanning set, parametric vector form, linear (in)dependence. Linear transformations: linear transformation, (co)domain, range and image, standard matrix, one-to-one and onto, singularity, determinant, elementary matrices. Subspaces and bases: subspace, column and null space, basis, coordinate system, dimension, rank. Eigenvalues and eigenvectors: eigenvalue, eigenvector, eigenspace, characteristic equation/polynomial, algebraic multiplicity, similarity, diagonalization and diagonalizability. Orthogonality: dot product, norm, distance, orthogonality, orthogonal complement, orthogonal set/basis, orthogonal projection, orthonormality, orthonormal basis, Gramm-Schmidt process, least squares problem/solution, orthogonal diagonalization, singular value/vector, singular value decomposition, Moore-Penrose inverse.

Teaching Methods

The course is spread over a period of seven weeks. Each week there will be two theoretical classes of 90 minutes each and two exercise classes of 90 minutes each.

Method of Assessment

There is a written exam at the end of the course.

Literature

Linear Algebra and its Applications, by David C. Lay, Steven R. Lay and Judi J. McDonald, global edition (sixth edition), Pearson or other material provided in the course.

Target Audience

Bachelor Artificial Intelligence (year 2)Bachelor Computer Science (year 2)