Dot product using a dual basis

University · ●○○○○ 10 pts

Let

B=\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_4\}

be a basis of

\mathbb{R}^4

with

\mathbf{e}_1=(1,0,0,0)

\mathbf{e}_2=(1,1,0,0)

\mathbf{e}_3=(1,1,1,0)

\mathbf{e}_4=(1,1,1,1)

. Its dual basis is

\mathbf{e}^1=(1,-1,0,0)

\mathbf{e}^2=(0,1,-1,0)

\mathbf{e}^3=(0,0,1,-1)

\mathbf{e}^4=(0,0,0,1)

. Let

\mathbf{u}=\mathbf{e}^1-\mathbf{e}^2+\mathbf{e}^3-\mathbf{e}^4

and

\mathbf{v}=(1,2,3,4)

, where vectors are written in the standard basis of

\mathbb{R}^4

. Find

\mathbf{u}\cdot\mathbf{v}

using the standard dot product.