Math101 · Independent Ontario learning library
10 · Subject library
Linear Algebra
Vectors, matrices, transformations, spaces, eigenvalues, and systems of equations.
Start with Basis →26 complete · 26 total lessons
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Complete subject sequence
Learn in a useful order.
Every topic below opens as a full lesson with interactive practice and a printable cheat sheet.
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01BasisA rigorous guide to bases, coordinates, change of representation, and proof of spanning plus independence.Full lesson + sheet02Cramer's RuleA precise determinant formula for square systems, with invertibility conditions and computational limits.Full lesson + sheet03DeterminantsA rigorous treatment of determinants as signed volume scaling, invertibility tests, and algebraic invariants.Full lesson + sheet04DiagonalizationA precise guide to diagonalizable matrices, eigenbases, powers, and the role of multiplicity.Full lesson + sheet05DimensionA rigorous explanation of dimension, basis size, rank–nullity, and finite-dimensional counting arguments.Full lesson + sheet06Echelon FormA precise guide to row echelon form, reduced echelon form, pivots, rank, and solution reading.Full lesson + sheet07EigenvaluesA rigorous guide to eigenvalues, characteristic polynomials, invariants, and their limits.Full lesson + sheet08EigenvectorsA precise guide to eigenspaces, nonzero-vector conditions, geometric multiplicity, and invariant directions.Full lesson + sheet09Gaussian EliminationA rigorous elimination algorithm for solving linear systems, detecting consistency, and controlling pivots.Full lesson + sheet10Gram-Schmidt ProcessA rigorous construction of orthogonal and orthonormal bases, with dependence checks and numerical cautions.Full lesson + sheet11Inverse MatrixA precise guide to matrix inverses, invertibility criteria, computation, and appropriate use.Full lesson + sheet12Kernel and RangeA rigorous treatment of null spaces, images, bases, dimensions, and the rank–nullity theorem.Full lesson + sheet13Least SquaresA rigorous guide to inconsistent systems, orthogonal residuals, normal equations, QR, and interpretation.Full lesson + sheet14Linear IndependenceA rigorous guide to linear independence, dependence relations, pivots, and proof strategies.Full lesson + sheet15Linear TransformationsA precise treatment of linear maps, matrices, kernels, images, composition, and basis dependence.Full lesson + sheet16MatricesA rigorous introduction to matrices as data arrays and representations of linear maps, with shape and entry notation.Full lesson + sheet17Matrix MultiplicationA precise explanation of matrix multiplication as composition, row–column pairing, and column combination.Full lesson + sheet18Matrix OperationsA rigorous guide to matrix addition, scalar multiplication, transpose, powers, and operation laws.Full lesson + sheet19Orthogonal ProjectionA precise treatment of projections onto lines and subspaces, residual orthogonality, and projection matrices.Full lesson + sheet20OrthogonalityA rigorous guide to inner products, orthogonal sets, complements, and orthonormal coordinates.Full lesson + sheet21Row ReductionA rigorous guide to elementary row operations, RREF, equivalence, rank, and subspace extraction.Full lesson + sheet22SpanA precise treatment of linear spans, membership, generating sets, and minimal bases.Full lesson + sheet23SubspacesA rigorous guide to the subspace test, bases, sums, intersections, and affine distinctions.Full lesson + sheet24Systems and MatricesA precise translation between linear systems, augmented matrices, vector equations, and solution geometry.Full lesson + sheet25Vector SpacesA rigorous foundation in vector spaces, axioms, examples, counterexamples, and coordinate-free reasoning.Full lesson + sheet26VectorsA precise introduction to vectors, components, magnitude, direction, linear combinations, and geometric operations.Full lesson + sheet
