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20.
(a) Is a composition of one-to-one linear transformations one-to-one? Justify your conclusion.
(b) Can the composition of a one-to-one linear transformation and a linear transformation that is not one-to-one be
one-to-one? Account for both possible orders of composition and justify your conclusion.
Show that defines a linear operator on but does not.
21.
22. is a linear transformation, then —that is, T maps the zero vector in into the
(a) Prove that if
zero vector in .
(b) The converse of this is not true. Find an example of a function that satisfies but is not a linear transformation.
Let l be the line in the -plane that passes through the origin and makes an angle with the positive x-axis, where .
23. Let be the linear operator that reflects each vector about l (see the accompanying figure).
(a) Use the method of Example 6 to find the standard matrix for T.
(b) Find the reflection of the vector about the line l through the origin that makes an angle of with the
positive x-axis.
Figure Ex-23 has exactly one solution for every vector w in
Prove: An matrix A is invertible if and only if the linear system
24. for which the system is consistent.
Indicate whether each statement is always true or sometimes false. Justify your answer by giving a
25. logical argument or a counterexample.
(a) If T maps into , and , then T is linear.
