Linear Transformations and Eigenvalues

Linear Transformations and Eigenvalues in Math Linear Algebra brings together calculators for understanding linear maps and their matrix representations, including applying a transformation to vectors, composing transformations, and using standard matrices for rotations, reflections, projections, and scalings. It also covers eigenvalues and eigenvectors through characteristic polynomials, eigenspaces, and core checks such as whether a matrix is diagonalizable.

This chapter works well for learners moving from basic matrix skills into more conceptual linear algebra, from early topics like interpreting a linear transformation as a function that preserves addition and scalar multiplication to advanced topics like change of basis, similarity, diagonalization, and the geometric meaning of eigenvectors as invariant directions. It’s suitable for university students and self-learners, while also offering depth for advanced users working with higher-dimensional matrices and computational verification.

Students and teachers can use these tools to practice and confirm results for eigenvalue problems, diagonalization steps, and transformation effects without getting stuck on algebra-heavy errors, while advanced users can quickly validate computations used in differential equations, stability analysis, Markov chains, PCA and data science, and computer graphics pipelines. This page helps connect theory to calculation, making eigenvalues and linear transformations easier to learn, apply, and trust.