Linear Algebra:

Let’s begin with the language of direction and magnitude. You’ll explore vectors, components, norms, and the geometry of lines and planes — discovering how algebra describes space with clarity and precision.

Matrices become powerful tools once you see their structure. You’ll learn matrix operations, inverses, and transformations — rotations, reflections, scalings — and understand how matrices reshape space and encode motion.

You’ll revisit systems from a new perspective — the matrix viewpoint. Using Gaussian elimination, echelon forms, and rank, you’ll see how solutions emerge from structure. Infinite solutions, no solution, unique solution — each case has geometric meaning.

More than just a number — determinants measure area, volume, and change under transformation. You’ll learn to compute determinants and interpret what they reveal about orientation, invertibility, and how a transformation stretches or compresses space.

This is where linear algebra becomes truly elegant. You’ll study vector spaces, subspaces, spans, independence, and bases — the framework that organizes all linear structures. These ideas show how mathematics generalizes patterns across dimensions.

You’ll explore the deep structure hidden inside matrices. Eigenvalues and eigenvectors reveal directions of pure stretching, and diagonalization simplifies complex transformations. These concepts are essential in physics, data science, and differential equations.

Using dot products and orthogonal bases, you’ll learn how to project vectors, decompose them, and understand least-squares problems. This is the mathematics behind approximations, statistics, and geometric reasoning.

Linear algebra is everywhere — in computer graphics, quantum mechanics, machine learning, population models, and more. We’ll explore examples tailored to your interests so you can see how vector structure shapes the world.

If you’re ready to go deeper, we can explore transformations in abstract vector spaces, spectral theory, norms and inner product spaces, or the linear algebra behind algorithms and machine learning.