Express the vectors u 1 ∗ , u 2 ∗ {\displaystyle {}u_{1}^{*},u_{2}^{*}} of the dual basis of the basis u 1 = ( 4 7 ) , u 2 = ( 6 − 1 ) {\displaystyle {}u_{1}={\begin{pmatrix}4\\7\end{pmatrix}},\,u_{2}={\begin{pmatrix}6\\-1\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}^{*}} .
