تجربة الجبر الخطي والمعادلات التفاضلية - Imam U

🔢 Part 1: Linear AlgebraThis section focuses on vector spaces, systems of equations, and structural transformations.Systems of Linear Equations: Solving linear systems using Gaussian elimination, Gauss-Jordan elimination, and row echelon forms.Matrices and Determinants: Matrix operations, inverse matrices, evaluating determinants, and applying Cramer’s Rule.Vector Spaces: Subspaces, linear independence, spanning sets, basis, and dimension of a vector space.Linear Transformations: Kernel (null space), range, matrix representation of a transformation, and the rank-nullity theorem.Eigenvalues and Eigenvectors: Finding characteristic polynomials, calculating eigenvalues/eigenvectors, and matrix diagonalization. 📉 Part 2: Differential Equations (ODEs)This section covers modeling and solving equations containing derivatives, focusing on first-order and higher-order Ordinary Differential Equations.First-Order Differential Equations:Separable variables equations.Homogeneous equations.Exact differential equations and integrating factors.Linear first-order equations (using the standard integrating factor \(I(x) = e^{\int P(x)dx}\)).Higher-Order Linear Differential Equations:Homogeneous equations with constant coefficients (using the auxiliary/characteristic equation).Non-homogeneous equations solved via the Method of Undetermined Coefficients or Variation of Parameters.Laplace Transforms (Often included): Using Laplace transforms to convert differential equations into algebraic problems for easier resolution, especially with initial value problems.