I am working through a Continuum Mechanics book for self study and came across the following question:
Let ${e_i}$ and ${e_i'}$ be two rectangular Cartesian base vectors. Show that if $e_i'=Q_{mi}e_m$ then $e_i = Q_{im}e_m'$.
I understand $Q$ to be an orthogonal tensor. Now the proof in the book goes as follows:
$e_i'=Q_{im}e_m \rightarrow e_i' \cdot e_j = Q_{mi}e_m \cdot e_j = Q_{mi} \delta_{mj} = Q_{ji} \rightarrow e_j = Q_{jm}e_m'$
Now my question is, how does that last implication follow above (i.e. how did we suddenly get $e_j = Q_{jm}e_m'$)?
If someone could explain that would be much appreciated.
As $\{e_i'\}$ vectors form an orthogonal basis, we can write: $$e_j = (e_j\cdot e_i')e_i' = Q_{ji}e_i'$$ as we know that $Q_{ji} = e_j\cdot e_i'$.