Page 593 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

be the linear transformations in Examples Example 11 and Example 12. Find for
(a)
(b)
(c)

32. (For Readers Who Have Studied Calculus) Let be the vector space of functions continuous on ,

and let be the transformation defined by

Is T a linear operator?

Indicate whether each statement is always true or sometimes false. Justify your answer by
33. giving a logical argument or a counterexample. In each part, V and W are vector spaces.

(a) If for all vectors and in V and all scalars

and , then T is a linear transformation.

(b) If is a nonzero vector in V, then there is exactly one linear transformation

such that .

(c) There is exactly one linear transformation for which
for all vectors and in V. defines a linear operator

(d) If is a nonzero vector in V, then the formula
on V.

If is a basis for a vector space V, how many different linear operators can
34. be created that map each vector in B back into B? Explain your reasoning.

Refer to Section 4.4. Are the transformations from to that correspond to linear
35. transformations from to necessarily linear transformations from to ?

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