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Math 211 Activities Workbook: Adapted from: Understanding Linear Algebra by David Austin

David Austin, Erica Bernstein

Section 3.1 matrix transformations geometry

Activity 3.1.1. Using matrix transformations to describe geometric operations.

Instructions.

The diagram below demonstrates the effect of a matrix transformation \(T\) on the plane. You may modify the matrix \(A=\begin{bmatrix} a \amp b \\ c \amp d \\ \end{bmatrix}\) defining \(T\) through the sliders at the top.
Since a matrix transformation takes a vector as input and produces a vector as output, we will show the inputs and outputs on separate sets of axes. In particular, the axes on the left represent the inputs while the axes on the right illustrate how input features are transformed by \(T\text{.}\)
Figure 3.1.1. The matrix transformation \(T\) transforms features shown on the left into features shown on the right.
For the following \(2\times2\) matrices \(A\text{,}\) use the diagram to study the effect of the corresponding matrix transformation \(T(\xvec) = A\xvec\text{.}\) For each transformation, describe the geometric effect the transformation has on the plane.
  1. \(A=\left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}\)
  2. \(A=\left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 2 \\ \end{array}\right]\text{.}\)
  3. \(A=\left[\begin{array}{rr} 0 \amp 1 \\ -1 \amp 0 \\ \end{array}\right]\text{.}\)
  4. \(A=\left[\begin{array}{rr} 1 \amp 1 \\ 0 \amp 1 \\ \end{array}\right]\text{.}\)
  5. \(A=\left[\begin{array}{rr} 1 \amp 0 \\ .5 \amp 1 \\ \end{array}\right]\text{.}\)
  6. \(A=\left[\begin{array}{rr} -1 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}\)
  7. \(A=\left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 0 \\ \end{array}\right]\text{.}\)
  8. \(A=\left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}\)
Solution.
  1. This transformation stretches by a factor of 2 in the horizontal direction.
  2. This transformation stretches by a factor of 2 uniformly in all directions.
  3. This is a \(90^\circ\) clockwise rotation.
  4. This transformation is called a horizontal shear; it leaves horiontal grid lines parallel to the \(x \) axis but vertical grid lines are slanted.
  5. This transformation is called a vertical shear; it leaves vertical grid lines parallel to the \(y \) axis but horizontal grid lines are slanted.
  6. This transformation reflects vectors in the vertical axis.
  7. This transformation is called a projection; it produces the shadow of the vector on the horizontal axis.
  8. This transformation is called the identity; it causes no change.
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