Linear Transformations

Compose Transformations

by an angle of degrees
across the x-axis
across the y-axis
in both directions by a factor of
horizontally by a factor of
vertically by a factor of
horizontally by a factor of
vertically by a factor of
onto the line through ( , )

Determinant: 1
☑ Invertible ☑ Orientation Preserving ☑ Area Preserving

A linear transformation \(T:V \to W\) is a mapping between vector spaces \(V\) and \(W\) that preserves addition and scalar multiplication. That is, for all \(v_1\) and \(v_2\) in \(V\),

\(T(v_1+v_2)=T(v_1)+T(v_2)\)
\(T(av_1)=aT(v_1)\)

For finite dimensional spaces \(V\) and \(W\) over a field \(F\), a linear transformation can be represented as an \( m \times n\) matrix \(M\), where \(m = \dim_F(V)\) and \(n = \dim_F(W)\). Thus, given a vector \(v \in V\), the result of applying the transformation to \(v\) is \(Mv\).

A transformation is invertible when its associated matrix is invertible; that is, when it has a nonzero determinant. The orientation of the images on the plane are preserved with the determinant is positive, and the area is preserved when the determinant is -1 or 1.

This applet illustrates the effects of applying various linear transformations to objects in \( \mathbb{R}^2 \). Choose a shape to apply transformations to, and zoom and in out using the slider. The \(2 \times 2\) transformation matrix can be entered directly, or you can choose one of the preset transformations listed. Choosing a preset transformation will update the transformation matrix automatically.

Check the "Combine Transformation" box to compose transformations. Transformations are composed by multiplying on the left by subsequent matrices. For instance, suppose \(R\) is a rotation matrix, and \(S\) is a matrix that scales the object. Then \(RS\) is a transformation that first scales the object, and then rotates it, while \(SR\) is a transformation that rotates the object, followed by scaling. To enter \(RS\), check the "Combine Transformation" box, then choose an amount to scale followed by the rotation. \(SR\) is entered in the opposite order - first rotate, then scale. Note that these transformations will generally not be equal, since matrix multiplication is not commutative.

An additional feature of the applet is the ability to see where each point \((x,y)\) of the object is sent by a transformation \(T\). Clicking and holding the mouse while moving over any point will reveal a pair of vectors \((x,y)\) and \(T(x,y)\). If these vectors are scalar multiples (they overlap), the vector is an eigenvector of the transformation.

If the orientation of the shape has been reversed due to a reflection, the color of the shape will change from blue to grey. The original position of the shape is displayed as a dotted grey outline.