Invariant Lines, Points And Matrices

"1. You have a matrix. Every matrix represents a transformation. e.g. matrix (0 1 1 0) is a reflection on the line y=x matrix (cosX -sinX sinX cosX) is a rotation anticlockwise about origin.

2. We can therefore call this matrix -> "TRANSFORMATION MATRIX" (a matrix that causes transformation)

3. For every single transformation matrix, there is an invariant line. Sometimes, for 1 transformation matrix, there is more than one invariant line.

4. For a given transformation matrix, If any POINT/coordinate of the cartesian plane falls on this specific invariant line, when the transformation matrix is applied to IT, the point will be mapped onto another point, which has coordinates that lie on the same invariant line.

5. SUMMARY: A SPECIFIC transformation matrix "has" 1(+) invariant line(s). when this transformation matrix is applied to a point (on this invariant line), the point will be mapped onto the same invariant line, but with different coordinates.

IF the NEW coordinates are the same as the coordinates BEFORE - it is an invariant point.

6. Hence, for a SPECIFIC transformation matrix any point on its` invariant line, will NEVER move from that invariant line even after that THAT SPECIFIC transformation matrix has been applied"