- (a)
- If and are both one-to-one, show that is one-to-one.
- (b)
- If and are both onto, show that is onto.
Challenge Problems for Chapter 10
Given linear transformations
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 \begin{equation*} \vec{a}+\vec{b} = \left \{ a_{n}+b_{n}\right \} _{n=1}^{\infty } \end{equation*}
and define scalar multiplication by \begin{equation*} c\vec{a}=\left \{ ca_{n}\right \} _{n=1}^{\infty }\text{ where }\vec{a} =\left \{ a_{n}\right \} _{n=1}^{\infty } \end{equation*}
Is this a vector space?
Let be the set of ordered pairs of complex numbers. Define addition and scalar
multiplication in the usual way. \begin{equation*} \left ( z,w\right ) +\left ( \hat{z},\hat{w}\right ) = \left ( z+\hat{z},w+ \hat{w}\right ) ,\ u\left ( z,w\right ) \equiv \left ( uz,uw\right ) \end{equation*}
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