Page 175 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 159

for all vectors v, vi, V2 in V and all scalars c in F. In short
the function T is required to act in a "linear" fashion on sums
and scalar multiples of vectors in V. In the case where T is a
linear transformation from V to V, we say that T is a linear
operator on V.
Of course we need some examples of linear transforma-
tions, but these are not hard to find.

Example 6.2.1
Let the function T : R 3 —> R 2 be defined by the rule

Thus T simply "forgets" the third entry of a vector. From
this definition it is obvious that T is a linear transformation.
Now recall from Chapter Four the geometrical interpreta-
tion of the column vector with entries a, b, c as the line segment
joining the origin to the point with coordinates (a, b, c). Then
the linear transformation T projects the line segment onto the
xy-plane. Consequently projection of a line in 3-dimensional
space which passes through the origin onto the xy-p\ane is a
2
linear transformation from R 3 to R .
The next example of a linear transformation is also of a
geometrical nature.

Example 6.2.2
Suppose that an anti-clockwise rotation through angle 9 about
the origin O is applied to the xy-plane. Since vectors in R 2
are represented by line segments in the plane drawn from the
2 2
origin, such a rotation determines a function T : R —> R ;
here the line segment representing T(X) is obtained by rotat-
ing the line segment that represents X.