Exercise 11.3 (The Dual Map)

Question

Suppose $V$ and $W$ are vector spaces and $A: V ightarrow W$ is a linear map. We define a linear map $A^*: W^* ightarrow V^*$, called the dual map or transpose of $\boldsymbol{A}$, by
$$
\left(A^* \omegaight)(v)=\omega(A v) \quad 	ext { for } \omega \in W^*, v \in V .
$$
	extbf{Exercise 11.3.} Show that $A^* \omega$ is actually a linear functional on $V$, and that $A^*$ is a linear map.

Elu Thingol · Accepted Answer

\begin{proof} We demonstrate the validity of both assertions.
\paragraph{$A^*$ is linear.} Let $c \in \mathbb{R}$ and let $\omega,\gamma:W 	o \mathbb{R}$ be two duals in $W^*$. We have
\begin{align*}
    \forall v \in V: \quad (A^*(c\omega+\gamma))(v) &= (c\omega+\gamma)(Av)\
    &= c\omega(Av) + \gamma(Av)\
    &= c(A^*\omega)(v) + (A^*\gamma)(v).
\end{align*}
In other words, $A^*(c\omega+\gamma) = c(A^*\omega) + A^*\gamma$.
\paragraph{$A*\omega$ is a linear functional.} To demonstrate that $A*\omega: V 	o \mathbb{R}$ is linear, pick $c \in \mathbb{R}$ and $v,w \in V$. Es gilt
\begin{align*}
    (A^*\omega)(cv + w) &= \omega(A(cv + w))\
    &= \omega(cAv + Aw) &	ext{Linearity von $A:V 	o W$}\
    &= c\omega(Av) + \omega(Aw) &	ext{Linearity von $\omega:W 	o \mathbb{R}$}\
    &= c(A^*\omega)(v) + (A^*\omega)(w).
\end{align*}
\end{proof}

Exercise 11.3 (The Dual Map)

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Exercise 11.3 (The Dual Map)

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