Linear Algebra- I - P Sattam

Systems of Linear Equations & Matrices:Row reduction techniques, Row Echelon Form (REF), and Reduced Row Echelon Form (RREF).Gaussian elimination and Gauss-Jordan elimination methods.Matrix operations (addition, scalar multiplication, multiplication, transposes).Inverting square matrices and checking for invertibility.Determinants:Evaluation methods (cofactor expansion and row operation properties).Invertibility criteria using determinants (\(\det(A) \neq 0\)).Cramer's Rule for solving specific systems.Vector Spaces:Axioms of a real vector space and common examples (\(\mathbb{R}^{n}\), matrices, polynomials).Subspaces, linear combinations, and the span of a set of vectors.Linear independence and dependence.Basis, dimension, and finding coordinates relative to a basis.Fundamental matrix subspaces (row space, column space, null space) and the Rank-Nullity Theorem.Linear Transformations:Definition, checking linearity, and standard matrix representations.Kernel (null space) and Image (range) of a transformation.Eigenvalues & Eigenvectors:Characteristic equations and solving for scalar values (λ).Finding eigenspaces and diagonalizing square matrices.