Undergraduate Linear Algebra

Summary

This course provides an introduction to the fundamental concepts of linear algebra, including systems of linear equations, matrix operations, determinants, vector spaces, eigenvalues, and eigenvectors. Emphasis is placed on both computational techniques and theoretical understanding, with applications in geometry, physics, computer science, and engineering. Students will develop problem-solving skills and gain insights into the structure and transformations of linear systems. The course also introduces linear independence, bases, dimension, and inner product spaces, preparing students for further studies in mathematics and related disciplines.

Content

• Sets and Maps
• Vector Spaces
• Dimension
• Linear Maps
• Matrix Calculus
• Determinants
• Systems of Linear Equations
• Euclidean Vector Spaces
• Eigenvalues
• The Principal Axes Transformation
• Classification of Matrices

Learning Prerequisites

• High-School level Math and maturity

Learning Outcomes

By the end of the course, the student must be able to:

• Apply methods such as Gaussian elimination and matrix inversion.
• Compute sums, products, inverses, and determinants of matrices.
• Define and analyze subspaces, bases, and dimensions.
• Determine kernels, ranges, and matrix representations.
• Solve characteristic equations and apply diagonalization techniques.
• Use dot products, orthogonality, and the Gram-Schmidt process.
• Demonstrate understanding of linear independence, span, and transformations.
• Solve problems in physics, engineering, computer science, and other fields.
• Utilize mathematical software for solving linear algebra problems.
• Construct logical arguments and proofs related to linear algebra concepts.

References

• Linear Algebra by Klaus Jänich
• Finite-Dimensional Vector Spaces by Paul R. Halmos
• Linear Algebra Done Right by Sheldon Axler