Linear Transformations and Their Properties

Non-linear transformations

Quadratic transformations

Linear transformations in two dimensions

Linear transformations in three dimensions

One-dimensional space

Three-dimensional space

Four-dimensional space

Two-dimensional space

They are difficult to calculate

They make the equations non-linear

They are unnecessary for defining linear transformations

They complicate the equations

Equation

Map

Graph

Matrix

It is negated

It is doubled

It is squared

It remains the same

uv = t(x, y) = (-x, -y)

uv = t(x, y) = (x, y)

uv = t(x, y) = (x, -y)

uv = t(x, y) = (y, x)

Points move in the opposite direction

Points move twice as far from the origin

Points move half as far from the origin

Points remain at the same distance from the origin

Rotating a point

Expanding a point

Reflecting a point

Pushing the top of a building to slant it

They make transformations non-linear

They are not significant

They ensure transformations are linear

They simplify calculations

As equations

As matrices

As vector fields

As graphs