Challenge Problems for Chapter 10

Given linear transformations

(a)
If and are both one-to-one, show that is one-to-one.
(b)
If and are both onto, show that is onto.
Let be a linear transformation.
(a)
If is one-to-one and for linear transformations , show that .
(b)
If is onto and for linear transformations , show that .
Consider functions defined on having values in . Explain how, if is the set of all such functions, can be considered as .
Let and be linear transformations.
(a)
If is one-to-one, show that is one-to-one and that .
(b)
If is onto, show that is onto and that .
Let denote the space of all functions (see Problem prb:10.21). If is defined by show that is an isomorphism.
Let denote the set of all real valued sequences. For two of these, define their sum to be given by and define scalar multiplication by Is this a vector space?
Let be the set of ordered pairs of complex numbers. Define addition and scalar multiplication in the usual way. Here the scalars are from . Show this is a vector space.

Bibliography

Several problems come from Chapter 9 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 469–535.

Several problems come from the end of Chapter 7 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 396.

2024-09-06 23:46:10