Summary Linear Algebra (First Year)

This document covers first-year university-level linear algebra. It features notes on each topic as well as practice problems done in class with answers. The units covered are: 1. Vectors, sets, lines, and planes (notation, dot product) 2. Systems of linear equations (matrices, Gauss-Jordan elimination, REF, RREF, types of matrices, types of solutions, matrix operations, matrix rank 3. Linear transformations (scaling, orthogonal projection, reflections, rotations, shear) 4. Composition and inversion of linear transformations (injective/surjective/bijective transformations, invertibility, composite transformations) 5. Determinant (absolute value of determinant, cross product, cofactor expansion, Cramer's rule) 6. Subspaces, span, linear dependence, and basis (subspaces, spans, images, kernels, linear relations, linear independence, linear dependence, basis) 7. Basis & coordinates (dimension, rank-nullity theorem, coordinates, change of coordinates matrix) 8. Orthogonal projection (orthogonal basis, orthonormal basis, orthogonal complement, orthogonal projection, Graham-Schmidt, QR factorization) 9. Orthogonal transformations and least squares (orthogonal transformations, transposes, least-square solution) 10. Eigenvalues and eigenvectors (diagonalizability, eigenvectors, eigenbasis, characteristic polynomial, algebraic multiplicity, eigenspace, geometric multiplicity, the spectral theorem)