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1.6 Linear Transformation
An operator denoted L will be called a linear function or linear operation if the following
relationship holds:
L [a + b] =  L (a) +  L (b)
where ,  are real numbers and a, b are vectors in the vector space. For real-valued continuous
functions and  and  scalars, the linear operation can be written as
L [f(t) + g(t)] =  L [f(t)] +  L [g(t)]
Operations such as differentiation, integration, Laplace and Fourier transforms are linear. The
d
derivative operator L =
is linear since
dt
d
df (t )
dg (t )
[f(t) + g(t)] = 
+
dt
dt
dt
Matrix multiplication of a linear combination of vectors is a linear operation since
A(x + y) = Ax + Ay
where L = A and  and  are scalars. In general, an operation that transforms a vector in Rn
(vector with n real components) to a vector in Rm is linear if and only if it coincides with
multiplication by some mn matrix.
The function (transformation) that converts a vector in Rn to a vector in Rm
f
Rn 
Rm
is said to have domain in Rn and range in Rm. When the transformation is linear, there exists a
unique mn matrix AT such that
L(x) = ATx
A subset S of a vector space V is a linear subspace if every linear combination of elements of S
is also in S. The linear subspace is usually called a subspace.
The rotation of a vector x through an angle  to become the vector x’ is a linear transformation
x’ = ATx
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where the transformation matrix AT is called the rotation matrix R. Figure 1.6 shows the rotation
of a two-dimensional vector with length r. We will determine the ration matrix R for this
transformation.
x2
(x',y')
(x,y)

a
x1
Figure 1.6 Rotation of a two-dimensional vector r
From the geometry, the coordinates of x’ in terms of the angle  and  are
x’ = rcos( + ) = [rcos()]cos()  [rsin()]sin()
y’ = rsin( + ) = [rsin()]cos() + [rcos()]sin()
The terms in the square bracket [rcos()] and [rsin()] are replaced by x and y to give
x’ = xcos()  ysin()
y’ = ycos() + xsin() = xsin() + ycos()
or
 x '  cos 
 y ' =  sin 
  
 sin  
cos  
x
 y
 
cos 
The transformation matrix for the plane rotation is then R = 
 sin 
angle  is zero the rotation matrix is simply the identity matrix I.
 sin  
. When the rotation
cos  
In three-dimensional rotations, the axis of rotation must be specified. The subscripts for a
rotation matrix will indicate the axis of rotation and the angle.
1. Rotation by an angle  about the x-axis:
0
1

Rx, = 0 cos 

0 sin 
0 
 sin  

cos  
2. Rotation by an angle  about the y-axis:
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 cos 
Ry, =  0

 sin 
0 sin  
1
0 

0 cos  
3. Rotation by an angle  about the z-axis:
cos 
Rz, =  sin 

 0
 sin 
cos 
0
0
0

1
The order in which the 3D rotations are performed is important since the 3D rotational matrices
do not commute in general. The rotation matrices are orthogonal matrices, so the columns (or
rows) must be orthonormal.
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