Page 176 - A Course in Linear Algebra with Applications

160 Chapter Six: Linear Transformationns

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To show that T is a linear operator on R , we suppose that
2
Y is another vector in R .

T(X+Y)

Referring to the diagram above, we know from the trian-
gle rule that X+Y is represented by the third side of the trian-
gle formed by the line segments representing X and Y. When
the rotation is applied to this triangle, the sides of the result-
ing triangle represent the vectors T(X), T(Y), T(X) + T(Y),
as shown in the diagram. The triangle rule then shows that
T(X + Y) = T(X)+T(Y).
In a similar way we can see from the geometrical inter-
2
pretation of scalar multiples in R that T(cX) = cT(X) for
any scalar c. It follows that T is a linear operator on R .
Example 6.2.3

Define T : D^a, b] —> Doo[a, b] to be differentiation, that is,

T(f(x)) = f'(x).

Here Doo[a,b} denotes the vector space of all functions of x
that are infinitely differentiable in the interval [a ,b]. Then
well-known facts from calculus guarantee that T is a linear
operator on D^a, b\.

This example can be generalized in a significant fashion
as follows. Let a±, a 2 ,..., a n be functions in D^a, b]. For any