\chapter{Dual spaces}


Dual spaces are the linear spaces formed by a space's linear forms (we will ensure these form linear spaces shortly).

Dual spaces are also fundamental in functional analysis and multilinear algebra (and consequently differential geometry).




\begin{definition}[Dual space]
Let $(V,F,+,\cdot)$ be a linear space over $F$, its \emph{dual space} $(V^{*},F,+,\cdot)$ is the linear space over $F$ formed by $V^{*}=\mathcal{L}(V,F)$ (all the linear forms AKA covectors on $V$)
\end{definition}

As usual, we verify that the dual space is indeed a linear space over $F$.


\begin{proposition}
Let $(V,F,+,\cdot)$ be a linear space spanned by n linearly independent vectors $\{\mathbf{b}_1 , \hdots, \mathbf{b}_n \}$. Then a basis for $V^*$ are the linear forms $\mathbf{b}^1 , \hdots , \mathbf{b}^n$ that obey the following.
	\[ \mathbf{b}^{i}(\mathbf{b}_i) = \delta_{ij} \]
\end{proposition}


Note that $\mathbf{b}^{i}$ are linear forms that are basis elements of our dual space; we typically represent covectrs with the same bold font as vectors, however we index covectors of a basis by superscripts rather than subscripts; This convention will prove very convenient in the study of tensors, since tensors will ultimately be a way of combining vectors and covectors.


Since we can use the linear space to construct a basis for its dual space with the same amount of basis elements, our dual space has the same dimension as the original space.

\begin{proposition}
Let $(V,F,+,\cdot)$ be a linear space, then its dual space has the same dimension as $V$.
\[\mathrm{dim}(V^*)=\mathrm{dim}(V) \]
\end{proposition}



An interesting physical interpretation due to our proposition is that dual linear spaces represent vectors by their projections onto basis vectors rather than linear combinations.


\subsection{Covariant transform}

\[ \mathbf{c}_i = \sum_{j} [\mathbf{P}]_{ji} \mathbf{b}_{j}\]
\[ [\mathbf{v}]_{B} = \mathbf{P} [\mathbf{v}]_{C}\]


\[ \mathbf{c}^i = \sum_{j} [\mathbf{P}]_{ij} \mathbf{b}^{j}\]
\[ [\mathbf{v}]^{C} = \mathbf{P} [\mathbf{v}]^{B}\]


Covectors transform covariantly with respect to the vector transition matrix.
Vectors transform contravariantly with respect to the vector transition matrix.

\section{Annihilators}


\section{Transpose of linear map}

\begin{definition}[Transpose linear map]
Let $(V,F,+,\cdot),(W,F,+,\cdot)$ be a linear spaces over $F$ and $f : V \to W$ be a linear map, then the \emph{transpose of $f$} is the linear map $f^{\intercal} : W^* \to V^*$ defined as the following. $f^{\intercal} (l) = l \circ f$.
\end{definition}