Rotating the Basis of the Vector Space

 

• $\boldsymbol{e}_{l'} = \boldsymbol{e}_{l}\Lambda^l{}_{l'}$
• $\boldsymbol{\omega}^{l'} = \Lambda^{l'}{}_{l}\boldsymbol{\omega}^l$
• 
  

  $$\delta^{j'}{}_{k'} \langle \boldsymbol{\omega}^{j'}, \boldsymbol{e}_{k'} \rangle = \... ...bda^{j'}{}_{j}\boldsymbol{\omega}^j, \boldsymbol{e}_{k}\Lambda^k{}_{k'} \rangle$$ =

  $$\Lambda^{j'}{}_{j} \Lambda^k{}_{k'} \langle \boldsymbol{\omega}^j, \boldsymbol{e}_{k} \rangle = \Lambda^{j'}{}_{j} \Lambda^k{}_{k'} \delta^j{}_k$$ =

  $$\Lambda^{j'}{}_{l} \Lambda^l{}_{k'}$$ = (3.52)

  
  
  hence $\left[\Lambda^{j'}{}_{l}\right] = \left[\Lambda^l{}_{k'}\right]^{-1}$
• For $\boldsymbol{\Lambda} \in \hbox{SO}(n)$ we have that $\boldsymbol{\Lambda}^{-1} = \boldsymbol{\Lambda}^{T}$.
• How does rotating the basis affect coefficients of vector a?
  

  $$a^l\boldsymbol{e}_l = a^l \boldsymbol{e}_{l'}\Lambda^{l'}{}_l$$ a =

  $$\left(\Lambda^{l'}{}_l a^l\right) \boldsymbol{e}_{l'}$$ =

  $$\Longrightarrow a^{l'} = \Lambda^{l'}{}_l a^l$$ (3.53)

  
  
  Observe that vector coefficients do not transform the same way as basis vectors. They transform like basis forms, i.e., ``in the opposite direction'' to the basis vectors. For this reason upper index objects are often called contravariant vectors.  
• How does rotating the basis affect coefficients of a form?
  

  $$\boldsymbol{\eta} \eta_l\boldsymbol{\omega}^l = \eta_l\Lambda^l{}_{l'}\boldsymbol{\omega}^{l'}$$ =

  $$\left(\eta_l\Lambda^l{}_{l'}\right) \boldsymbol{\omega}^{l'}$$ =

  $$\Longrightarrow \eta_{l'} = \eta_l\Lambda^l{}_{l'}$$ (3.54)

  
  
  Form coefficients, as you see, transform like basis vectors. For this reason lower index objects are often called covariant vectors.  
• Now consider a tensor F:
  

  $$F^i{}_j\boldsymbol{e}_i\otimes\boldsymbol{\omega}^j = F^i{}_j\boldsymbol{e}_{i'}\Lambda^{i'}{}_i\otimes \Lambda^j{}_{j'}\boldsymbol{\omega}^{j'}$$ F =

  $$\left(\Lambda^{i'}{}_i F^i{}_j \Lambda^j{}_{j'}\right) \boldsymbol{e}_{i'}\otimes\boldsymbol{\omega}^{j'}$$ =

  $$\Longrightarrow F^{i'}{}_{j'} = \Lambda^{i'}{}_i F^i{}_j \Lambda^j{}_{j'}$$ (3.55)

  
  
  This formula merely confirms what we have already learnt about forms and vectors.