Express the vectors u 1 ∗ , u 2 ∗ {\displaystyle {}u_{1}^{*},u_{2}^{*}} of the dual basis of the basis u 1 = ( 1 3 ) , u 2 = ( 2 − 5 ) {\displaystyle {}u_{1}={\begin{pmatrix}1\\3\end{pmatrix}},\,u_{2}={\begin{pmatrix}2\\-5\end{pmatrix}}} in R 2 {\displaystyle {}\mathbb {R} ^{2}} as linear combinations with respect to the standard dual basis e 1 ∗ , e 2 ∗ {\displaystyle {}e_{1}^{*},e_{2}^{*}} .
