Plane Transformations

This section is devoted to illustration of linear transformations on the plane. Then each such transformation T: ℝ² ↦ ℝ² is identified by a square 2×2 matrix A, and vise versa. Plane transformations can be classified as reflections, contractions/expansions, shears, and projections The following tables give appropriate terminologies for such transformations.


  1. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that first reflects points through the horizontal x1-axis and then reflects points through the line x1 = x2.
  2. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that first rotates points through -3π/4 radian (clockwise) and then reflects points the vertical x2-axis.
  3. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that maps i=(1,0) into 2i-3j but leavs the vector j=(0,1) unchanged.
  4. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that rotates points (about the origin) through 3π/2 radians (counterclockwise).
  6. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that rotates points (about the origin) through -π/4 radians (clockwise).
  7. If \( {\bf A} = \begin{bmatrix} 1&2 \\ -1&-2 \end{bmatrix} , \) find two matrices B ≠ C such that AB = AC.

  1. Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 236ff