Dual Vector Space

definitionvector-spaces

Definition

Given a vector space $V$, we have $V^{\prime }=\mathcal{L}(V,\mathbb{F})$ is the dual space.

Dual Basis

Given a basis $v_1,\dots ,v_n$ of $V$, we can form the dual basis, ${\phi }_1,\dots ,{\phi }_n\in V^{\prime }$ defined by,

$${\phi }_j(v_k)=\{\begin{array}{ll}1& j=k\\ 0& j\ne k\end{array}$$

The elements ${\phi }_j$ are a basis of $V^{\prime }$ if $V$ is finite-dimensional.

Dual Transformation

Given a linear map $T\in \mathcal{L}(V,W)$, the dual of $T$ is $T^{\prime }\in \mathcal{L}(W^{\prime },V^{\prime })$ defined by $T^{\prime }(\phi )=\phi \circ T$ for $\phi \in W^{\prime }$.

Examples