Linear Algebra Concept Notes
These standalone notes give compact explanations of selected linear algebra ideas. They are organized by concept and include references, equations and links to related GraphMath chapters when available.
Change of Basis
Coordinate mapping between bases, with the vector separated from its coordinate descriptions.
Open change of basis noteRow Reduction
Gaussian elimination interpreted as multiplication by elementary matrices, with column transformations and row-normal views.
Open row reduction noteLinear Equations
Unique solution, infinitely many solutions and no solution interpreted geometrically through planes in 3D.
Open linear equations noteDeterminant
Signed area and volume visualized through elementary matrices, row operations and controlled geometric transformations.
Open determinant noteDeterminant formulas
Permutation and cofactor formulas visualized as signed products, pivots, submatrices and determinant calculation structure.
Open determinant formulas noteDeterminant visualization
Orientation, determinant sign and cofactor signs interpreted through projected cyclic order and related visualizations.
Open determinant visualization noteCramer's rule
Determinant ratios for solving a square full-rank system, with a geometric interpretation of signed volumes and solution coordinates.
Open Cramer's rule noteWhy Cramer’s Rule Works
Geometric derivation of Cramer's rule in R³ and Rⁿ, explaining why column replacement isolates one solution coordinate.
Open Cramer's rule geometric derivation noteShape and Rank
A compact classification of the main matrix shape-and-rank cases, showing one-to-one, onto, solution behavior, determinant, inverse and pseudoinverse consequences.
Open shape and rank summary table noteProjection Formula
Orthogonal projection onto a vector and onto a column space derived from the defining condition that the residual is perpendicular to the projection subspace.
Open projection formula derivation noteLeast Squares Projection
Least squares as orthogonal projection in ℝ³ for three observations and two parameters.
Open least squares noteQR Decomposition
QR as an upper triangular transformation followed by an orthogonal transformation in 2D and 3D.
Open QR decomposition noteGram-Schmidt
Stepwise orthogonalization and normalization as a geometric construction of an orthonormal basis.
Open Gram-Schmidt noteGivens Rotation Algorithm
Plane rotations used one at a time to eliminate selected matrix entries and build QR factorization.
Open Givens rotation noteHouseholder Reflection
Reflection across a plane used to align a column with a coordinate direction in QR factorization.
Open Householder reflection noteRodrigues Rotation Formula
Rotation around a fixed 3D axis by separating the axial and perpendicular components of a vector.
Open Rodrigues rotation noteThis concept-note collection is intended to grow as more GraphMath topics are added.
Related work: GraphMath Linear Algebra
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