About Lec 1 | MIT 18.06 Linear Algebra, Spring 2005

Solving Ax=0 via Reduced Row Echelon Form to Find the Null Space Basis Linear Independence of Basis Vectors in Euclidean Space Positive Definite Matrices and Quadratic Forms in Linear Algebra for Minimizing Functions Cramer's Rule: A Geometric Interpretation via Determinants and Volume Scaling Linear Transformations Preserving Addition and Scaling Operations Linear Algebra: Row Exchanges and Permutation Matrices in Factorization Incidence Matrices and Kirchhoff's Laws for Electrical Networks Markov Matrix Steady States and Fourier Series as Orthogonal Projections Dimension and Basis for Matrix Spaces from Symmetric Matrices to Small World Graphs Eigenvalues of Symmetric, Skew-Symmetric, and Orthogonal Matrices in Linear Algebra Linear Systems of Equations in Linear Algebra Linear Algebra: Matrix Multiplication Rules and Gaussian Elimination to Find Inverse Matrices LU Factorization for Solving Linear Systems Determinant Properties in Linear Algebra Matrix Representation of Linear Transformations via Basis Mapping Four Fundamental Subspaces in Linear Algebra Linear Algebra: Orthogonal Complements and Null Spaces Eigenvalues and Eigenvectors Definitions in Linear Algebra Linear Algebra: Column Space and Nullspace of Matrices Singular Value Decomposition via Eigenvalue Decomposition of A^TA and AA^T Quiz Review of Projections, Eigenvalues, and Determinants in Linear Algebra Solving Inconsistent Linear Systems Using Least Squares Matrix Similarity: A = PBP^-1 and Invariant Spectral Properties Eigenvalues and Eigenvectors for Symmetric Matrices Projection Matrix Formulas in Linear Algebra Left and Right Inverses and the Pseudoinverse Rectangular Matrix Subspaces and Rank in Linear Algebra Gram-Schmidt Process and QR Factorization Complex Numbers and Hermitian Matrices in Linear Algebra via the Fast Fourier Transform (FFT) Solution Sets of Ax=b and Ax=0: Affine Subspaces vs. the Null Space Matrix Determinant Calculation Methods in Linear Algebra Solving Ax=b Using Gaussian Elimination and Back Substitution Matrix Diagonalization: A = PDP^-1 via Eigenvectors and Eigenvalues Solving Linear Differential Equations via the Matrix Exponential exp(At)