• EDUCATIONAL SIMULATION LAB

Linear Algebra
Simulation Lab

Explore core concepts through live, interactive visualizations. Drag vectors, adjust matrices, and see mathematics in action.

MODULE 1

Vectors & Linear Combinations

2D Plane

A vector is an arrow with both magnitude and direction. In linear algebra we care about how vectors combine.

LINEAR COMBINATION

Two vectors u and v form a basis for ℝ² if they are linearly independent (not parallel). Then any vector w can be written uniquely as w = a·u + b·v.

Current linear combo:

Coefficient a (for u)

1.00

Coefficient b (for v)

0.70

MODULE 2

Matrix Transformations

Linear Maps on ℝ²

Transformation Matrix A

a₁₁

1.20

a₁₂

0.40

a₂₁

-0.30

a₂₂

0.90

Determinant 1.20

MODULE 3

Eigenvalues & Eigenvectors

Invariant Directions

An eigenvector of a matrix A is a nonzero vector v such that A v = λ v. The direction stays the same (or reverses); only the length changes by the factor |λ|.

Eigenvalue λ₁

—

Eigenvalue λ₂

—

Drag any vector on the canvas to test it. Eigenvectors will stay on their line after transformation.

MODULE 4

Determinants & Volume Scaling

The determinant of a 2×2 matrix tells you the signed area of the parallelogram formed by its column vectors. It is also the factor by which areas are scaled under the linear transformation.

Signed Area (det)

1.08

Absolute Area

1.08

Drag the tips of the two column vectors to change the matrix

MODULE 5

Solving Linear Systems

A system of linear equations Ax = b can be visualized geometrically as the intersection of lines (in 2D) or planes (in 3D).

Unique solution — lines intersect at one point

∞

Infinite solutions — lines coincide

No solution — lines are parallel and distinct

Equation 1: a x + b y = c

Equation 2: d x + e y = f