Interactive lessons

Matrix Algebra
from first principles

Each lesson is an interactive tool you can fiddle with — drag sliders, watch transformations happen in real time, build geometric intuition before touching a formula. This series grows from what a matrix is into the full language of linear transformations, systems, and decompositions.

Foundations

— 3 of 3 complete
2 −1 4 3 0 7 2 × 3 rows columns
Foundations — 01
What Is a Matrix?
Rows, columns, entries, and notation. Build matrices by clicking cells, name their dimensions, and see how vectors are just special cases.
interactive · 2D · editable →
1 3 2 4 + 5 0 −1 2 = 6 3 1 6 entry-by-entry scalar ×
Foundations — 02
Addition & Scalar Multiplication
Edit matrices entry-by-entry and watch sums form live. Drag a scalar slider and see every entry scale together — the geometry of stretching a grid.
interactive · 2D · sliders →
1 2 3 4 5 6 7 8 9 × a b c d e f = row · col dot product step-by-step (m×n)(n×p)=(m×p)
Foundations — 03
Matrix Multiplication
Click any result cell and watch the matching row and column light up. Step through the dot product, understand why order matters, and see which dimension pairs must agree.
interactive · 2D · step-through →

Transformations & Systems

— 3 of 3 complete
e₁ e₂ T(x) = Ax ▶ animate
Transformations — 04
Linear Transformations
Drag the columns of a 2×2 matrix and watch the entire grid deform live. See rotation, scaling, shear, and reflection as different choices of basis vectors.
interactive · 2D · animated →
2 1 −1 8 −3 −1 2 −11 −2 1 2 −3 R₂ + ³⁄₂R₁ R₃ + R₁ row ops pivots
Systems — 05
Gaussian Elimination
Walk through row reduction step by step on your own system. Each operation is explained, pivots glow, and the echelon staircase emerges naturally.
interactive · step-through · editable →
det(A) = area a = [a₁, a₂] b = [b₁, b₂] ad − bc ▶ drag area
Transformations — 06
The Determinant
Drag two column vectors and watch the parallelogram they span resize. The determinant is that area — signed. See it flip when vectors cross.
interactive · 2D · draggable →

Structure & Decomposition

— 3 of 3 complete
A A⁻¹ 1 0 0 1 ⋯ A A⁻¹ = I undone A A⁻¹ = I
Structure — 07
The Inverse Matrix
Apply A, then apply A⁻¹, and arrive back at the identity. Explore when inverses exist, how they relate to the determinant, and how to compute them via row reduction.
interactive · 2D · animated →
v₁ λ₁v₁ v₂ Av = λv ▶ rotate λ · eigenval
Structure — 08
Eigenvalues & Eigenvectors
Rotate a matrix transformation and find the special directions that only stretch — never rotate. See how the characteristic polynomial gives you the eigenvalues.
interactive · 2D · animated →
Col(A) Null(A) rank + nullity = n 1 + 1 = 2 rank nullspace dim theorem
Structure — 09
Rank, Nullspace & Dimension
Build a matrix and watch its column space and null space appear as geometric subspaces. The rank-nullity theorem becomes a visible identity: rank + nullity = n.
interactive · 2D · subspaces →