Exercises 4.3 Exercises
Exercise Group.
In the following exercises verify that the given map is an \(F\)-linear transformation.
2.
\(T\colon\Q(\sqrt{2})\to\Q(\sqrt{2})\) given by \(a+b\sqrt{2}\mapsto 2a\text{.}\) Here \(\Q(\sqrt{2})\) is considered as a vector space over \(\Q\text{.}\)3.
Let \(C(\R)\) be the set of all real-valued continuous functions on \(\R\text{.}\) Consider
\begin{equation*}
T\colon C(\R)\to C(\R)\quad\text{given by}\quad f\mapsto Tf\text{ where }Tf \text{ is a map defined by }x\mapsto\int_0^xf(t)dt.
\end{equation*}
4.
Let \(V\) be a finite-dimensional vector space over a field \(F\text{.}\) Let \(\{v_1,v_2,\ldots,v_n\}\) be its basis. For each \(v_i\) define \(f_i\in V^*\) as follows.
\begin{equation*}
f_i(v_j)=\begin{cases}1\amp\text{if }i=j\\0\amp\text{if }i\neq j\end{cases}.
\end{equation*}
Consider \(T\colon V\to V^*\) defined by
\begin{equation*}
\sum\alpha_iv_i\mapsto\sum\alpha_if_i.
\end{equation*}
5.
Let \(V,W\) be vector spaces over a field \(F\text{.}\) Show that if \(\varphi\colon V\to W\) is an \(F\)-linear map then there exists an \(F\)-linear map \(\varphi^*\colon W^*\to V^*\text{.}\)
Hint.
If \(f\colon W\to F\) is a linear functional then, we define \(\varphi^*(f)=f\circ\varphi\text{.}\) In diagrammatic terms (refer Section 4.9):
\begin{equation*}
\begin{CD}V@>\varphi>>W\\@.@VVfV\\@.F\end{CD}
\end{equation*}
