About What's the big idea of Linear Algebra? **Course Intro**

Gram-Schmidt Orthogonalization Linear Transformations in Linear Algebra Finding a Basis for the Null Space and Column Space via Row Reduction Diagonalizing Matrices with Complex Eigenvalues and Eigenvectors Mathematics and visual arts The Rank-Nullity Theorem for Matrices Elementary Matrices: Representing Row Operations via Matrix Multiplication Matrix Representation Theorem: Linear Transformations Determined by Standard Basis Images Orthogonal Basis Coefficient Formula via Dot Product Projections Span and Basis of the Polynomial Vector Space P_n Dimension of a Vector Subspace: Invariance of Basis Vector Count Matrix-Vector Multiplication via Row-Column Dot Products and the Column Space Eigenvalues and Eigenvectors Definitions in Linear Algebra Mathematics in general Testing Vector Span Membership via Gaussian Elimination Matrix-Vector Multiplication: Ax = b as a Linear Combination of Columns Homogeneous Linear Systems: Trivial Solutions, Closure, and Solution Decomposition Matrix Multiplication in Linear Algebra Matrix Similarity: A = PBP^-1 and Invariant Spectral Properties Diagonalization as Geometric Scaling in the Eigenbasis Vector Angle and Orthogonality via the Dot Product Mathematics for nonmathematicians (engineering, social sciences, etc.) Change of Basis via the Invertible Transition Matrix 2x2 Matrix Inverse Formula Orthogonal Projections in Linear Algebra Diagonal Matrices: Rank, Eigenvalues, and Similarity Invariants Determining Linear Independence vs Linear Dependence Gram-Schmidt Process: Recursive Projection Formula Walkthrough Solution Sets of Ax=b and Ax=0: Affine Subspaces vs. the Null Space Integer Lattices Generated by Linear Combinations of a Basis Equivalence of Matrix Transformations and Linear Transformations Linear Independence and Dependence of Vectors Determinant Calculation via Laplace Cofactor Expansion Along Any Row or Column Dimensions of the Null Space and Column Space via the Rank-Nullity Theorem Matrix Diagonalization: A = PDP^-1 via Eigenvectors and Eigenvalues