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 transformation
is represented by the matrix
Consider the effect ofon the line
This may be represented by the vector
sends
to
Hence
and
Rearranging the first of these gives
Substitute this into the second to obtain