- Introduction to matrices
- Adding and subtracting matrices
- Multiplying matrices
- 2 × 2 Matrices and linear transformations
- Determinants of 2 × 2 matrices
- Inverses of 2 × 2 matrices
- Invariant points and lines in 2 dimensions
- 3 × 3 Matrices and linear transformations
- Determinants of 3 × 3 matrices
- Inverses of 3 × 3 matrices
- Matrices and simultaneous equations
Part 7: Invariant points and lines in 2 dimensions
An invariant point under a transformation is a point that maps to itself. As noted in part 4, linear transformations map the origin to the origin, so the origin is always an invariant point under a linear transformation.
An invariant line is a line that maps to itself. To be precise, every point on the invariant line maps to a point on the line itself. Note that the point needn’t map to itself.
A a line of invariant points is a line where every point every point on the line maps to itself. Any line of invariant points is therefore an invariant line, but an invariant line is not necessarily always a line of invariant points.
Use this applet to see invariant points, invariant lines, and lines of invariant points for three examples of linear transformations.