Linear Algebra (1) - NU

1. Systems of Linear Equations and MatricesLinear Systems: Solving systems of linear equations using Gaussian Elimination and Gauss-Jordan Elimination.Matrix Operations: Rules for matrix addition, scalar multiplication, matrix multiplication, and finding the transpose.Matrix Inverses: Computing the inverse of a matrix using row operations and properties of invertible matrices.2. DeterminantsEvaluation: Calculating determinants using cofactor expansion and row reduction techniques.Properties: Analyzing how row operations alter determinants and understanding the multiplicative property (\(\det(AB) = \det(A)\det(B)\)).Applications: Utilizing Cramer’s Rule to solve linear systems and finding matrix inverses via the adjoint matrix.3. Vector SpacesGeneral Vector Spaces: Verifying the 10 axioms required to define a formal vector space over the real numbers.Subspaces: Identifying subsets that are closed under addition and scalar multiplication.Linear Independence: Determining whether a set of vectors is linearly independent or dependent.Basis and Dimension: Finding a minimal spanning set (basis) for a vector space and identifying its dimension.Four Fundamental Subspaces: Exploring the row space, column space, and null space of a matrix.4. Eigenvalues and EigenvectorsCharacteristic Equation: Solving \(\det(\lambda I - A) = 0\) to find eigenvalues.Eigenspaces: Computing eigenvectors corresponding to specific eigenvalues by finding the null space.Diagonalization: Determining if a matrix A can be factored into P D P⁻¹ where D is a diagonal matrix.