Linear Algebra

Systems & Matrices

1. Gaussian Eliminationrow echelon form, reduced form, free variables, rank3–6 h
2. Matrix Algebraoperations, inverses, transpose, elementary matrices, LU decomposition3–6 h
3. Determinantscofactor expansion, properties, Cramer's rule, geometric interpretation3–6 h

Vector Spaces

1. Vector Spaces & Subspacesaxioms, span, linear independence, examples (R^n, polynomials, functions)3–6 h
2. Basis & Dimensionbasis characterization, dimension theorem, coordinate vectors3–6 h
3. Row Space, Column Space & Nullspacerank-nullity theorem, fundamental subspaces3–6 h

Linear Transformations

1. Linear Maps & Matricesmatrix representation, change of basis, kernel and image3–6 h
2. Eigenvalues & Eigenvectorscharacteristic polynomial, eigenspaces, diagonalization3–6 h
3. Similarity & Jordan Form Introinvariant subspaces, generalized eigenvectors3–6 h

Inner Products & Applications

1. Inner Product Spacesnorms, angles, orthogonality, Gram-Schmidt process3–6 h
2. Orthogonal Projections & Least Squaresbest approximation, regression connection, QR decomposition3–6 h
3. Spectral Theoremsymmetric (self-adjoint) operators, orthogonal diagonalization, singular values intro3–6 h
4. Linear Algebra ApplicationsMarkov chains, differential equations systems, Google PageRank, image processing3–6 h