Linear algebra deals with vectors, matrices, and linear transformations. It began with solving systems of linear equations but grew into a language for geometry, physics, and data science. A vector is a list of numbers; a matrix is a rectangular array. Operations on them follow rules that generalize arithmetic.

The subject studies vector spaces, bases, dimension, eigenvalues, and eigenvectors. These concepts describe rotations, projections, and scaling. Solving Ax = b is central: it asks for the vector x that a matrix A maps to b.

Applications are everywhere. Computer graphics use matrices to rotate and translate objects. Machine learning relies on matrix operations for training models. Quantum mechanics uses vector spaces to describe states. Engineers solve structural problems with linear systems. Linear algebra is often the first advanced mathematics course students take, and for good reason.

• Studies vectors, matrices, and linear transformations
• Includes vector spaces, eigenvalues, and bases
• Central to graphics, machine learning, and physics
• Solving Ax = b is a core problem