Math · Linear Algebra

Linear Transformations

A matrix describes a transformation of space. Move a vector and watch the same rule reshape both the vector and the coordinate grid.

Concept

Vectors are the objects being transformed. Matrices describe the calculation. A linear transformation shows what that calculation does geometrically: $T (x) = A x$ .

Equation

$x \to A x$

Manipulate

Drag the orange arrow, or focus its endpoint and use the arrow keys. The blue grid and arrow show $A x$ ; the fine gray grid remains as a reference.

Basis vectors reveal the rule

$e_{1} = (1, 0)$ and $e_{2} = (0, 1)$ are the standard basis vectors. The transformed basis arrows are the columns of A.

Calculate $A x$

Determinant: area scaling

The shaded parallelogram is the transformed unit square. Its area makes determinant tangible without needing a separate determinant lesson.

Composition and order

One transformation after another is still a matrix transformation. Changing the order changes which rule acts first.

Observe

Scale stretches, reflection reverses a direction, shear slants the grid, rotation turns it, and projection collapses the plane onto a line. When $\det (A) = 0$ , the collapse loses information.

Computing, data & machine learning

Graphics uses matrices to transform coordinates and shapes. Data science uses transformations to change representations. A common ML layer is $y = W x + b$ : a learned matrix W transforms the input vector x before later operations.

Eigenvectors

Most vectors change direction under a transformation. But some keep pointing along the same line and only change magnitude: $A v = λ v$ . Which vectors keep their direction?