Modern interactive lesson

Linear Transformations in R^n

Explore linearity, standard matrices, kernels, images, and how matrices move vectors and grids.

MathJax formulasLive calculatorsCanvas visualizationQuick checks

Big idea: a matrix is a motion rule

A linear transformation $T:\mathbb R^n\to\mathbb R^m$ preserves linear combinations:

$$T(cu+dv)=cT(u)+dT(v).$$

Algebra

Every matrix transformation $T(x)=Ax$ is linear.

Geometry

Linear maps can rotate, scale, shear, reflect, project, and collapse space, but they always send $0$ to $0$.

Testing linearity

Fast test: if $T(0)\ne0$, then $T$ is not linear.

Linearity tester

sample check

The standard matrix

If $T:\mathbb R^n\to\mathbb R^m$ is linear, the columns of its standard matrix are the images of the standard basis vectors:

$$A=\begin{bmatrix}|&&|\\T(e_1)&\cdots&T(e_n)\\|&&|\end{bmatrix}.$$

Build a 2D standard matrix

columns = images

Transformation studio

live grid

Choose a transformation and watch how it moves the coordinate grid and a test vector.

Gray grid = original grid. Colored grid = transformed grid. Green arrow = input vector. Yellow arrow = output vector.

Kernel, image, one-to-one, onto

Kernel

$\ker(T)=\{x:T(x)=0\}=\operatorname{Null}(A)$.

Image

$\operatorname{im}(T)=\{T(x):x\in\mathbb R^n\}=\operatorname{Col}(A)$.

2 by 2 kernel/image inspector

rank-nullity

Chapter summary

Linear

Preserves addition and scalar multiplication.

Matrix

Columns are images of basis vectors.

Information

Kernel measures information lost; image measures possible outputs.

Quick check

Which function cannot be linear?