1.11.1 Tensor calculus

Problem: In \( \mathbb{R}^{3} \) given a basis \( \mathcal{E}=\left\{e_{1}, e_{2}, e_{3}\right\} \) three linear forms by coordinate notation in \( \mathcal{E} \), forming a basis \( \mathcal{T}=\left\{f_{1}(x), f_{2}(x), f_{3}(x)\right\} \) of a dual space: \( e_{1}=(1 ; 3 ; 2), e_{2}=(1 ; 0 ; 1), e_{3}=(2 ;-1 ; 1) ; \) \( f_{1}(x)=2 x_{1}-x_{2}+2 x_{3} \), \( f_{2}(x)=x_{1}+x_{2}+x_{3}, f_{3}(x)=3 x_{1}+x_{3} \). 1) Find a basis in \( \mathbb{R}^{3} \), reciprocal with basis \( \mathcal{T} \). 2) Write in basis \( \varepsilon \) the basis \( e^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right\} \), reciprocal with the standard basis in \( \mathbb{R}^{3} \). 3) Find a basis reciprocal with \( \varepsilon \).