Matrix Transformations – Linear Algebra

1. Functions and Transformations

In standard algebra, a function takes a single number as an input and outputs a new number: f(x) = y.

In Linear Algebra, we scale this concept up. A Transformation (often denoted as T) is a mathematical rule that takes an entire Vector as an input and assigns it to a new Vector output. We write this as T(x) = b.

You can think of a transformation as a machine that takes a point in space, moves it according to specific rules, and drops it in a new location.

2. Matrix Transformations

The magic of linear algebra is that every linear transformation can be perfectly represented by a Matrix. By multiplying a coordinate vector by a transformation matrix, we can scale, rotate, reflect, and shear objects in space.

Try the interactive visualizer below. Watch how changing the values in the 2×2 Matrix (A) physically alters the “F” shape on the 2D Cartesian plane.

3. Properties of Matrix Transformations

For a transformation to be considered a mathematically valid Linear Transformation, it must rigidly follow two specific properties. If a matrix represents the transformation, these properties are automatically guaranteed.