Matrices
1.1 Definitions
Definition of a matrix, element notation, size, transpose, and special matrix types: row, column, square, diagonal, identity, symmetric, anti-symmetric, triangular, null.
1.2 Operations with Matrices
Matrix equality, addition, scalar multiplication, and matrix multiplication — definitions, worked examples, and the transpose rule (AB)ᵀ = BᵀAᵀ.
1.4 Determinants
Determinant of 2×2 and 3×3 matrices, minors, cofactors, Laplace expansion, and key properties including det(AB) = det(A)det(B).
1.7.4 Gaussian Elimination
Solving linear systems AX = B using the Gaussian elimination algorithm — augmented matrix, elementary row operations, row echelon form, unique/infinite/no solutions, overdetermined and underdetermined systems.
1.5 Matrix Inversion
Definition of the matrix inverse, inverse of 2×2 via formula, inverse of n×n via adjoint matrix, singularity condition, and laws of matrix algebra.
1.8 Rank of a Matrix
Rank of a matrix via row echelon form, full rank, the rank-nullity theorem, null space and homogeneous systems, and solvability of AX = B.
1.7.3 Cramer's Rule
Cramer's rule for solving n linear equations in n unknowns using determinants — 2×2 and 3×3 examples including electric circuit current problems.
Vectors
2.1 The Geometric Approach
Definition of vectors, magnitude, direction, triangle rule for addition, Cartesian components, unit vectors i j k, position vectors, and basic vector operations.
2.3 Scalar (Dot) Product
Scalar (dot) product definition and component form, angle between vectors, projection, cross product determinant form, right-hand rule, scalar and vector triple products.
2.2 Vector Spaces
Vector space axioms, linear independence, span, basis, Gram-Schmidt orthogonalization process, orthogonal and orthonormal basis.
Linear Geometry
3.1 Equation of a Straight Line in Three Dimensions
Parametric and Cartesian equations of lines in 3D, direction vectors, distance between skew lines, angle between lines, intersection of lines, distance from point to line.
3.2 Equation of a Plane
Normal vector equation of a plane, scalar form, distance from point to plane and from origin, angle between planes, line-plane intersection, three-point plane, parametric form.
Linear Transformations
Linear Transformations
Definition of linear transformations, standard matrix representation, 2D transformations (rotation, reflection, scaling, shear), composition, invertible transformations.
Eigenvalues and Eigenvectors
Eigenvalue equation Ax=λx, characteristic polynomial, finding eigenvalues and eigenvectors, diagonalization, geometric and algebraic multiplicity, applications.
