The Image of a Line Under a Transformation Represented by a Matrix

Any transformationrepresented by a matrixwhose entries are numbers is linear. The matrix will send a line to a line. The transformed line will in general be rotated and scaled, but it will still be a line. For example, suppose a transformationis represented by the matrix

Consider the effect ofon the lineThe line may be represented by the vectorsendsto


Rearranging the first of these givesSubstitute this into the second to obtain

The labelsandare arbitrary, so we can write

The property of the transformation that sends a line to a line is a reflection of the fact that the entries in the matrix are constants, so the thatandare linear functions ofcan then be eliminated betweenandby simple substitution. The original and transformed lines are shown below