Matrices as transformations of the plane

Problem types

• Matrices as transformations

Builds on

• Operations on matrices
• Vectors and their components
• Inverse of a matrix · Useful idea

  The inverse of a transformation matrix undoes the transformation.

• Which matrix rules hold · Useful idea

  Matrix laws such as order of multiplication decide how a sequence of transformations composes.

• A system written as a matrix equation · Useful idea

  Solving AX = B asks which point the matrix A sends to B, so the inverse undoes the transformation.

• Motion described by a parameter · Useful idea

  A parametric circle is rotated or stretched by applying a matrix to the coordinate pair at each parameter value.

• Proofs about parallelograms · Useful idea

  A matrix maps the unit square to a parallelogram, and the area of that parallelogram is the determinant.

• What a transformation keeps · Useful idea

  Matrices that carry out rigid motions keep lengths and angles, the invariants of the motions.

Leads to

None listed.

Related skills

• The complex plane · Related idea

  Multiplying by i rotates a point 90 degrees, as a rotation matrix does to the coordinates.

• Data arranged in a matrix · Related idea

  Points stored as the columns of a matrix are transformed all at once by multiplying with another matrix.

• Polar and Cartesian coordinates · Related idea

  Adding an angle to theta rotates a point about the origin, which a rotation matrix does to the coordinates.

• Shifts, stretches, and reflections of graphs · Related idea

  Shifts, stretches and reflections of a graph are transformations, and matrices carry out such transformations on points.

• Choosing a periodic model · Related idea

  A rotation matrix with entries cosine and sine turns each point by an angle, the same periodic pattern as a sinusoid.

• How a dilation changes a figure · Related idea

  A scale matrix is a dilation about the origin, which multiplies every length by the scale factor and every area by its square.

• Constructing the image of a figure · Related idea

  Multiplying matrices composes transformations: the product AB applies B first and then A. This is the matrix form of carrying one figure to another by a sequence of moves.