Linear Algebra is the study of vectors, matrices, and linear transformations. It is one of the most widely applicable branches of mathematics.

Key Topics

• Vectors and Vector Spaces — Operations, subspaces, basis, and dimension.
• Matrices — Addition, multiplication, transpose, and inverse.
• Systems of Linear Equations — Gaussian elimination and row echelon form.
• Linear Transformations — Functions between vector spaces that preserve addition and scalar multiplication.
• Determinants — A scalar that encodes information about a matrix: \(\det(A)\).
• Eigenvalues and Eigenvectors — Directions that are stretched (not rotated) by a transformation: \(A\mathbf{v} = \lambda\mathbf{v}\).
• Orthogonality — Orthogonal bases, Gram-Schmidt process, and least squares.
• Symmetric Matrices and SVD — Spectral decomposition and singular value decomposition.

Why It Matters

Linear algebra is the backbone of computer graphics, data science, quantum mechanics, and machine learning. Every time you rotate a 3D model, compress an image, or train a neural network, linear algebra is at work.