In linear algebra, given a vector space ${\displaystyle V}$ with a basis ${\displaystyle B}$ of vectors indexed by an index set ${\displaystyle I}$ (the cardinality of ${\displaystyle I}$ is the dimension of ${\displaystyle V}$), the dual set of ${\displaystyle B}$ is a set ${\displaystyle B^{*}}$ of vectors in the dual space ${\displaystyle V^{*}}$ with the same index set ${\displaystyle I}$ such that ${\displaystyle B}$ and ${\displaystyle B^{*}}$ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ${\displaystyle V^{*}}$. If it does span ${\displaystyle V^{*}}$, then ${\displaystyle B^{*}}$ is called the dual basis or reciprocal basis for the basis ${\displaystyle B}$.

Denoting the indexed vector sets as ${\displaystyle B=\{v_{i}\}_{i\in I}}$ and ${\displaystyle B^{*}=\{v^{i}\}_{i\in I}}$, being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes are equal, and equal to 0 otherwise. Symbolically, evaluating a dual vector in ${\displaystyle V^{*}}$ on a vector in the original space ${\displaystyle V}$:

where ${\displaystyle \delta _{j}^{i}}$ is the Kronecker delta symbol.

To perform operations with a vector, we must have a straightforward method of calculating its components. In a Cartesian frame the necessary operation is the dot product of the vector and the base vector. For example,

where ${\displaystyle \{\mathbf {i} _{1},\mathbf {i} _{2},\mathbf {i} _{3}\}}$ is the basis in a Cartesian frame. The components of ${\displaystyle \mathbf {x} }$ can be found by

However, in a non-Cartesian frame, we do not necessarily have ${\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=0}$ for all ${\displaystyle i\neq j}$. However, it is always possible to find vectors ${\displaystyle \mathbf {e} ^{i}}$ in the dual space such that

The equality holds when the ${\displaystyle \mathbf {e} ^{i}}$s are the dual basis of ${\displaystyle \mathbf {e} _{i}}$s. Notice the difference in position of the index ${\displaystyle i}$.

The dual set always exists and gives an injection from V into V^∗, namely the mapping that sends v_i to vⁱ. This says, in particular, that the dual space has dimension greater or equal to that of V.