Linear Algebra Visual Explanations
These pages collect visual explanations of selected linear algebra ideas. Each page is centered on diagrams or animations, with enough accompanying text to connect the visual structure to the algebra.
| Preview | Visual explanation | Related chapter |
|---|---|---|
|
Change of Basis — Coordinate Mapping Visualized Coordinate mapping between bases, with the geometric vector separated from its coordinate descriptions. | Linear transformations |
|
Row Reduction as Multiplication by Elementary Matrices Column view and row-normal-plane view of Gaussian elimination as multiplication by elementary matrices. | Row reduction |
|
Linear Systems in 3D — Solution Types Visualized Unique solution, no solution and infinitely many solutions interpreted as configurations of planes in three dimensions. | Linear Equations |
|
Determinant — Signed Area and Volume Visualized Signed area in 2D and signed volume in 3D interpreted through elementary matrices and their effect on the unit square or cube. | Determinant |
|
Determinant Sign Visualization Orientation and determinant sign read from projected cyclic order in 3D, with a related cofactor expansion visualization. | Determinant sign |
|
Cramer's Rule — Determinant Ratios Visualized Animated 3D interpretation of determinant ratios, signed volumes and the signs of solution components. | Cramer's rule |
|
Least Squares Projection in R³ Visualized Least squares in ℝ³ as orthogonal projection onto a model subspace, with the residual perpendicular to the fit. | Projection |
|
QR Decomposition — Geometric Interpretation in 2D and 3D Separation of an upper triangular deformation from an orthogonal transformation in 2D and 3D QR examples. | QRF, Gram-Schmidt, part 1 |
|
Gram-Schmidt Orthogonalization — Stepwise Visualization in 2D and 3D Stepwise correction and normalization of columns as an orthonormal basis is constructed. | QRF, Gram-Schmidt, part 3 |
|
Givens Rotation Algorithm — Geometric Interpretation in 3D Plane rotations zero selected entries one at a time while building the triangular factor in QR factorization. | Givens rotations |
|
Householder Reflection Visualized Reflection used to align a column with a coordinate direction while building QR factorization. | — |
|
Visualizing Rodrigues Rotation Formula Rotation around a fixed 3D axis by separating axial and perpendicular components of a vector. | 3D rotation |
This visual collection is intended to grow as more GraphMath topics are added.
Related work: GraphMath Linear Algebra
Contact: info@graphmath.com