A Level Maths Notes: FP4 – The Image of a Line Under the Transformation Represented by a Matrix
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 ofon
the line
The
line may be represented by the vector
sends
to
Hence
and
Rearranging the first of these gives
Substitute
this into the second to obtain
The labels
and
are
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 of
can
then be eliminated betweenandby
simple substitution. The original and transformed lines are shown
below
