Ballinteer Institute
Linear
Transformations :
Transformation This is an operation in the Coordinate plane or vector space which
maps 
Examples of transformations
are (1) Reflections in the X axis or a Central Symmetry in (0,0)ie 
Other transformations can be
written as follows 
which can be
expressed as 
is called the
transformation matrix of the transformation
The Course requires that we find
the image of points by a given transformation
If f is a transformation such that
ie 
if a(0,0) b (2,1)
c(3,4) Find f(a),f(b),f(c) The simplist way is to Multiply the
transformation matrix by each of
the points ie 
We are often asked
to write x and y in terms of 
. To find x and y we know 
premultiply both
sides of the equation by the inverse of the transformation matrix . The inverse of 
this gives
. We can use this relationship to find the image of a Line or
a Line segment by the transformation
just replace x and y by their images in( x1, y 1)
Ex : find the image of the
line 3x + 4y + 5 = 0 by f the result is
It is a requirement
that we show the inverse to be true ie -5(3x+y)+9(2x+y)+5=0 = 3x+ 4y+5=0 .
Example if H : ax+by+c = 0 and K : ax+ by+d = 0 are 2 parallel lines Show f(H) and f(K) are parallel
both have
slope
therefore f(H) is parallel to f(K)
Example if L : px +qy +r =
0 and N: qx - py+ +t = 0 are 2 perpendicular lines investigate if f(L) is
perpendicular to f(N) .
The slope of f(L) is -p+2q
the slope of f(N) is -q +2p the product of the slopes is not equal to
- 1
-p+3q -q-3p fL) is not perpendicular to f(N)
To write a Line segment in
parametric form we use an idea from vectors
If x is any point on a line segment [ab] wher o is the origin then 
|
|
This enables to write any
line segment [ab] in parametric form if t = 0 x = a, if t = 1 x = b if x 
To find the image of
a line segment by a transformation just multiply the transformation matrix by
the Line segment written in parametric form .
NOTE 
EXAMPLEa( 1,2 ) , b =
(4,1) write [ab] in parametric form . We know if p is a point on [ab] then p =
tb + (1-t)a gives x = 4t + (1-t)1= x = 3t+1,
y = 1t+ (1-t)2 = y = 2 -t So the image of [ab]by f is
x = 8t +5, y = 5t+4
Properties Of linear
Transformations
If f is a Linear
transformation then
(1) f(0) = 0 . The origin
is it's own image .
(2) If a and b are two
points then f(a+b) = f(a) +f(b) .
(3) The image of a line is
a line , the image of a line segment is a line segment.
(4) the image of a pair of
parallel lines is a pair of parallel lines .
(5) If L and K are 2 lines
and 
.
(6) The Image of a
parallelogram is a parallelogram .
(7)The images of two
perpendicular lines are not necessarily perpendicular .
(8) If A is the area of a
triangle abc, the area of f(abc) is k A where k is the
determinent of the transformation matrix .
(9) When asked to find the
image of a line in the form ax+by+c = 0 by a transformation noot only must you
find it's image but you must also transform it back to verify that all of the
original line is mapped onto the image . If the line is in parametric form
there is no need for the inverse transformation .
Properties of 2x2 Matrices
(1) If 
(2)
The trace of A is a
+ d, (3) The determinent of
A is ad -bc .
(4)The Characteristic
equation of A is the
Quadratic 
(5)the Inverse of Matrix A
is 
=
(6)Note If A and B are two
Matrices Show 
(7) A diagonal Matrix is a
Matrix of the form K=
note 
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