Skip to main content

Section 7.2 Rotations, reflections, projections and dilations

Subsection 7.2.1 Transformations \(T\colon \R^2\to\R^2\)

We wish to consider some transformations that are geometrically inspired. The following examples from \(\R^2\) will be useful as we study linear transformations.

Example 7.2.1. A rotation in the plane.

We start with a point \(\vec x\) in the plane and rotate it through an angle \(\theta\) counterclockwise around the origin. This new point is \(L(\vec x)\text{.}\)

Figure 7.2.2. The vector \(\vec x\) rotated to \(L(\vec x)\) though an angle \(\theta\)
Example 7.2.3. A reflection in the plane.

We start with a point \(\vec x\) in the plane and reflect in across the line with equation \(y=x\text{.}\) This new point is \(L(\vec x)\text{.}\)

Figure 7.2.4. The reflection of the vector \(\vec x\) by the line \(y=x\)
Example 7.2.5. A projection in the plane.

We start with a point \(\vec x\) in the plane and drop a perpendicular to the line with equation \(y=x\text{.}\) This new point is \(L(\vec x)\text{.}\)

Figure 7.2.6.
Example 7.2.7. A dilation in the plane.

We start with a real number \(s\gt 0\text{.}\) For a point \(\vec x\text{,}\) let \(L(\vec x)=s\vec x\text{.}\) If \(\vec x\not=\vec0\text{,}\) then \(L(\vec x)\) is on the line joining \(\vec x\) and \(\vec 0\text{.}\) The distance from \(\vec x\) to \(\vec 0\) has been stretched by a factor of \(s\) to get the distance from \(L(\vec x)\) to \(\vec 0\text{.}\)

Figure 7.2.8.