Any transformation represented by a matrix whose 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 of
on the line
This may be represented by the vector
sends
to
Henceand
Rearranging the first of these givesSubstitute this into the second to obtain