صفحات العروض بالإنجليزية في الوقت الحالي؛ أما الدرس المرتبط بكل عرض فهو بالعربية بالكامل.
كيف تقرأ هذه الصفحات. كل صفحة تبيّن أي أرقامها محسوبة، وأيها مأخوذ من شكل منشور، وأيها توضيحي. وقد أُنتج بعضها بمساعدة الذكاء الاصطناعي وتذكر ذلك على الصفحة نفسها. هي أدوات تعليمية، وليست قياسات لعتاد حقيقي.
التجارب العملية
العروض التفاعلية
Vector Playground
من درس ما هو الجبر الخطي؟
Drag two vectors and watch their sum, dot product, angle and projection update live; the determinant decides whether their span is a line or the whole plane.
Matrix as a transformation of the plane
من درس التحويلات الخطية
Drag a vector and watch the basis and the determinant follow. Eigenvectors emerge as the directions that do not turn.
Eigenvalue Explorer
من درس مسألة القيم الذاتية
Set a 2×2 matrix and watch its characteristic polynomial, eigenvalues and eigenvectors compute live; rotate a unit vector and see its image land parallel to itself only along an eigen-line, while a negative discriminant means a complex pair and no real eigen-direction.
Linear System Solver
من درس إعداد الأنظمة الخطية
Type a 2×2 or 3×3 system A x = b and watch Gaussian elimination with partial pivoting run to row echelon form, the determinant appear as the product of the pivots, and the system classified as one solution, none, or infinitely many — with the two lines drawn for the 2×2 case.
SVD and low-rank approximation
من درس تحليل القيم الشاذة
Move a rank slider and watch an image rebuild itself one singular value at a time, with the retained energy computed at every rank.
Orthogonality & Projections
Orthonormalize two vectors with Gram–Schmidt, project a vector onto the span with its orthogonal residual, and fit a line by least squares — every basis vector, projection, residual and coefficient computed from the normal equations, with the residual shown orthogonal to the column space.
Fourier Series Builder
من درس الضرب الداخلي
Rebuild a periodic waveform (square, sawtooth, triangle, pulse) from its Fourier harmonics, with each coefficient computed as an inner product of the function with a sine or cosine — the series as an orthogonal projection. Watch the RMS error fall as terms are added, the orthogonality table sit at zero off the diagonal, and the ~9% Gibbs overshoot at a jump narrow but never shrink.
Numerical Experiments
من درس الفاصلة العائمة والتكييف
Set a 2×2 or 3×3 matrix and watch the numerics: the condition number κ = σmax/σmin computed by power and inverse-power iteration on AᵀA, the image ellipse whose axis ratio is κ, Jacobi and Gauss–Seidel with their convergence trace and the spectral radius that decides convergence, and how ill-conditioning amplifies a perturbation of b within the κ bound.