Linear Algebra
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
16 lessons
- Vectors in n DimensionsExtend vectors from the plane to lists of n numbers: componentwise addition and scaling, length and the dot product in ℝⁿ, and the standard basis vectors.
- Linear Combinations and SpanBuild new vectors by scaling and adding old ones, and describe everything reachable that way: the span of a set, a line, a plane or all of ℝⁿ through the origin.
- Row Reduction and Echelon FormSolve any linear system by row reducing its augmented matrix: three row operations, echelon form, pivots, and the reduced form that reads off the answer.
- Solution Sets of Linear SystemsRead the shape of a solution set from the pivots: none, exactly one, or infinitely many, written in parametric vector form as a particular solution plus directions.
- Linear IndependenceA set of vectors is independent when the only combination giving zero is the trivial one. Test it by row reduction, and read independence from the pivot columns.
- Matrix Multiplication and Ax as a Combination of ColumnsRead Ax as a linear combination of the columns of A, multiply matrices of any compatible size, and see the product AB as doing B first and then A.
- Linear Transformations and Their MatricesA function on vectors is linear when it respects sums and scalar multiples. Every such function is multiplication by a matrix, built column by column from e₁ to eₙ.
- The Inverse of a MatrixFind A⁻¹ by row reducing [A | I], see why that works, use the inverse to solve Ax = b, and meet the Invertible Matrix Theorem, which ties the course together.
- Determinants: Cofactor Expansion, Row Operations and AreaCompute determinants by cofactor expansion or by row reducing to triangular form, see how row operations change them, and read det A as an area or volume factor.
- Subspaces: Column Space and Null SpaceMeet subspaces, the sets closed under sums and scalar multiples, and the two every matrix carries: its column space in ℝᵐ and its null space in ℝⁿ, with a basis for each.
- Basis, Dimension and the Rank TheoremA basis spans a subspace with no redundancy, and every basis has the same size: the dimension. The rank theorem splits a matrix's columns between Col A and Nul A.
- Eigenvalues and EigenvectorsFind the directions a matrix only stretches: solve det(A − λI) = 0 for the eigenvalues, then the null space of A − λI for each eigenvector.
- Diagonalization and Powers of a MatrixWrite A = PDP⁻¹ with eigenvectors in P and eigenvalues in D, use it to compute Aᵏ in one step, and see where a Markov chain settles in the long run.
- Orthogonal Projections and Orthogonal BasesWith an orthogonal basis, coordinates come from dot products alone. Project a vector onto a line or subspace, and see why the projection is the closest point.
- The Gram–Schmidt Process and QR FactorizationTurn any basis into an orthogonal one by subtracting projections one vector at a time, normalize to get an orthonormal basis, and package the result as A = QR.
- Least Squares and the Line of Best FitWhen Ax = b has no solution, find the x that comes closest: solve the normal equations AᵀAx = Aᵀb, and use them to fit a line to data.