Dual Spaces

Abstract

In considering the elements of the vector space $\mathbb{R}^{d}$ as column vectors, the vector space $(\mathbb{R}^{d})^{t}$ of row vectors is obviously related to $\mathbb{R}^{d}$, but is not necessarily identical to it. What, therefore, is one to make of row vectors and of the transpose ξ ↦ ξ ^t operation applied to column vectors ξ? The usual product of matrices and vectors indicates that each ξ ^t is a linear transformation $\mathbb{R}^{d} \rightarrow \mathbb{R}$ via ξ ^t(η) = ξ ^t η, for (column) vectors $\eta \in \mathbb{R}^{d}$.