Linear Algebra

Course Description

MAS3105 – Linear Algebra is an upper-division mathematics course covering vector spaces, matrices, linear transformations, and the theory connecting them. It is required for mathematics majors and is a standard requirement or prerequisite in engineering, physics, computer science, data science, and economics programs.

Linear algebra occupies an unusual position in the mathematics curriculum. It is the first course most students take that is genuinely about abstraction and proof rather than computation, and that transition is the course's real difficulty. Students who have succeeded through calculus by learning procedures often struggle here, where the question shifts from "compute this" to "prove that this must be true."

Content begins concretely with systems of linear equations, Gaussian elimination, and matrix algebra — material that feels like an extension of prior coursework. It then moves into the abstract core: vector spaces and subspaces, linear independence, basis and dimension, linear transformations and their matrix representations, the four fundamental subspaces, rank and nullity, and the Rank-Nullity Theorem.

The latter portion covers determinants, eigenvalues and eigenvectors, diagonalization, and inner product spaces including orthogonality, the Gram-Schmidt process, and orthogonal projection. Eigenvalue theory is where the course pays off — it underlies principal component analysis, vibration analysis, Markov chains, quantum mechanics, and Google's original PageRank algorithm.

The applications are genuinely broad, and modern versions often make them explicit: linear algebra is the mathematical substrate of computer graphics, machine learning, structural engineering, econometrics, and signal processing.

Offered at approximately 17 Florida institutions including the State University System (FAMU, FAU, FGCU, FIU, FSU, UCF, UF, UNF, USF, UWF, Florida Poly) and state colleges offering upper-division mathematics.

Learning Outcomes

Required Outcomes

• Solve systems of linear equations using Gaussian and Gauss-Jordan elimination.
• Perform matrix operations including multiplication, transposition, and inversion.
• Determine whether a matrix is invertible and compute the inverse.
• Compute and interpret determinants and apply their properties.
• Verify whether a set with operations constitutes a vector space or subspace.
• Determine linear independence and dependence of a set of vectors.
• Find a basis for a vector space and determine its dimension.
• Identify the four fundamental subspaces of a matrix and compute bases for each.
• Apply the Rank-Nullity Theorem.
• Determine whether a mapping is a linear transformation and find its matrix representation.
• Perform change of basis and compute coordinate vectors.
• Compute eigenvalues and eigenvectors and determine whether a matrix is diagonalizable.
• Diagonalize a matrix and apply diagonalization to compute matrix powers.
• Apply inner products, orthogonality, and the Gram-Schmidt process.
• Construct clear mathematical proofs of results in linear algebra.

Optional Outcomes

• Apply least squares approximation to overdetermined systems.
• Describe singular value decomposition and its applications.
• Apply linear algebra to Markov chains and steady-state analysis.
• Use computational tools such as MATLAB, Python/NumPy, or Octave for matrix computation.
• Describe applications in computer graphics, data analysis, or engineering.
• Describe complex vector spaces and Hermitian matrices.

Major Topics

Required Topics

• Systems of linear equations — Gaussian elimination, row echelon form, and solution sets.
• Matrix algebra — operations, inverses, elementary matrices, and LU factorization.
• Determinants — computation, properties, cofactor expansion, and Cramer's rule.
• Vector spaces — axioms, subspaces, and standard examples.
• Linear independence, basis, and dimension — spanning sets and coordinate systems.
• The four fundamental subspaces — column space, null space, row space, and left null space.
• Rank and nullity — the Rank-Nullity Theorem and its consequences.
• Linear transformations — definition, kernel and image, and matrix representation.
• Change of basis — coordinate vectors and similarity.
• Eigenvalues and eigenvectors — characteristic polynomial, eigenspaces, and multiplicity.
• Diagonalization — conditions, procedure, and applications to matrix powers.
• Inner product spaces — inner products, norms, and orthogonality.
• Orthogonalization — Gram-Schmidt, orthogonal projection, and orthogonal matrices.
• Proof technique — constructing and writing rigorous arguments.

Optional Topics

• Least squares and regression.
• Singular value decomposition.
• Markov chains and steady states.
• Computational linear algebra with MATLAB, Python, or Octave.
• Applications in graphics, data science, and engineering.
• Complex vector spaces and spectral theory.

Resources & Tools

• Linear Algebra and Its Applications (Lay, Lay & McDonald), Pearson — among the most widely adopted texts.
• Elementary Linear Algebra (Anton & Rorres), Wiley — the other dominant option.
• Introduction to Linear Algebra (Strang), Wellesley-Cambridge — used where programs emphasize conceptual understanding; Strang's MIT OpenCourseWare lectures are freely available and widely recommended.
• Linear Algebra Done Right (Axler), Springer — for proof-focused sections.
• Computational tools — MATLAB, Python with NumPy, Octave, or Wolfram Alpha for checking work and exploring larger systems.
• 3Blue1Brown's Essence of Linear Algebra — a freely available video series widely used by students for geometric intuition.
• Graphing calculators with matrix capability, where permitted.

Career Pathways

Linear algebra is foundational infrastructure rather than an occupational course, but it gates a great deal:

• Data Scientist and Machine Learning Engineer — linear algebra is the mathematical basis of nearly every method in the field.
• Engineer (all disciplines) — structural analysis, control systems, and signal processing all rest on it.
• Statistician and Actuary (SOC 15-2041, 15-2011) — regression and multivariate methods are linear algebra.
• Computer Graphics and Game Developer — transformations, projections, and rendering.
• Operations Research Analyst (SOC 15-2031) — optimization and linear programming.
• Mathematics Teacher (SOC 25-2031) — required for Florida mathematics certification pathways.
• Graduate study — assumed knowledge in mathematics, physics, engineering, economics, and computer science.

Florida employers drawing on this background include aerospace and defense on the Space Coast and at Lockheed Martin, Northrop Grumman, and L3Harris; simulation and modeling firms in Orlando; and the growing data and analytics sector in Tampa and Miami.

Special Information

Credit and prerequisite variation — check both before registering

This course varies more than most across Florida, in two ways that matter. Credit value: the University of North Florida lists MAS3105 at 4 credits, while Florida A&M, Florida SouthWestern, and Miami Dade list it at 3. Prerequisite: UNF and FAMU require MAC2312 (Calculus II), while Florida SouthWestern requires MAC2313 (Calculus III). A student planning transfer should confirm both, since a 3-versus-4 credit difference affects degree audits and the calculus prerequisite affects when the course can be taken.

Related course numbers are not interchangeable

Florida also carries MAS3106 (Linear Algebra, often the proof-intensive version for mathematics majors) and MAS4105 (Linear Algebra, upper-division at some universities). SCNS equivalency applies to the same number at the same level, not across numbers. Students should confirm which number their degree program requires — taking MAS3105 when the program requires MAS4105 is a real and avoidable problem.

This is usually a student's first proof course, and that is the difficulty

Worth setting expectations plainly. Calculus rewards procedural fluency; linear algebra asks students to work with definitions, construct arguments, and reason about objects that have no picture. Students commonly report that the first half feels manageable and the material becomes hard the moment vector spaces are defined abstractly. The productive response is to engage with definitions precisely — writing them out, testing them against examples and non-examples — rather than looking for procedures to memorize. Many students benefit from working problems in groups, since articulating an argument aloud exposes gaps that silent reading does not.

Position in the curriculum

Typically taken in the sophomore or junior year after the calculus sequence. It is prerequisite to differential equations at some institutions, and to abstract algebra, numerical analysis, and most upper-division applied mathematics. For data science and machine learning pathways it is effectively foundational.

Gordon Rule and general education

MAS3105 is an upper-division mathematics course. It does not satisfy Florida general education core mathematics requirements, which are met at the 1000–2000 level.