Page 323 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(b) If is a one-to-one linear transformation, then there are no distinct vectors u

and v in such that .

(c) If is a linear operator, and if for some vector x, then is an

eigenvalue of T.

(d) If T maps into , and if for all scalars and and
for all vectors u and v in , then T is linear.

Indicate whether each statement is always true, sometimes true, or always false.
26.

(a) If is a linear transformation and , then T is one-to-one.

(b) If is a linear transformation and , then T is one-to-one.

(c) If is a linear transformation and , then T is one-to-one.

Let A be an matrix such that , and let be multiplication by A.
27.

(a) What can you say about the range of the linear operator T? Give an example that illustrates
your conclusion.

(b) What can you say about the number of vectors that T maps into 0?

In each part, make a conjecture about the eigenvectors and eigenvalues of the matrix A
28. corresponding to the given transformation by considering the geometric properties of multiplication

by A. Confirm each of your conjectures with computations.

(a) Reflection about the line .
(b) Contraction by a factor of .

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