Challenge Problems for Chapter 10

Given linear transformations

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1.
If \(T_1\) and \(T_2\) are both one-to-one, show that \(T\) is one-to-one.
2.
If \(T_1\) and \(T_2\) are both onto, show that \(T\) is onto.
Let \(T:V\rightarrow W\) be a linear transformation.
1.
If \(T\) is one-to-one and \(TR_1=TR_2\) for linear transformations \(R_1,R_2:U\rightarrow V\), show that \(R_1=R_2\).
2.
If \(T\) is onto and \(S_1T=S_2T\) for linear transformations \(S_1,S_2:W\rightarrow U\), show that \(S_1=S_2\).
Consider functions defined on \(\left \{ 1,2,\cdots ,n\right \} \) having values in \(\mathbb {R}\). Explain how, if \(V\) is the set of all such functions, \(V\) can be considered as \(\mathbb {R}^{n}\).
Let \(f\left ( i\right ) \) be the \(i^{th}\) component of a vector \( \vec {x}\in \mathbb {R}^{n}\). Thus a typical element in \(\mathbb {R}^{n}\) is \( \left ( f\left ( 1\right ) ,\cdots ,f\left ( n\right ) \right ) \).
Let \(T:v\rightarrow U\) and \(S:\rightarrow W\) be linear transformations.
1.
If \(ST\) is one-to-one, show that \(T\) is one-to-one and that \(\dim V\leq \dim U\).
2.
If \(ST\) is onto, show that \(S\) is onto and that \(\dim W\leq \dim U\).
Let \(\mathbb {D}_n\) denote the space of all functions \(f:\{1, 2, \dots , n\}\rightarrow \RR ^n\) (see Problem ). If \(T:\mathbb {D}_n\rightarrow \RR ^n\) is defined by
\[T(f)=(f(1), f(2), \dots , f(n))\]
show that \(T\) is an isomorphism.
Let \(\mathbb {R}^{\mathbb {N}}\) denote the set of all real valued sequences. For \(\vec {a}\equiv \left \{ a_{n}\right \} _{n=1}^{\infty },\vec {b}\equiv \left \{ b_{n}\right \} _{n=1}^{\infty }\) 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 \(\mathbb {C}^{2}\) 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 \(\mathbb {C}\). 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.