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$$ \begin{pmatrix}x'_{1}\\ x'_{2}\\ x'_{3} \end{pmatrix} = \begin{pmatrix} \overline{e}'_1. \overline{e}_1 & \overline{e}'_1. \overline{e}_2 & \overline{e}'_1. \overline{e}_3\\ \overline{e}'_2. \overline{e}_1 & \overline{e}'_2. \overline{e}_2 & \overline{e}'_2. \overline{e}_1 \\ \overline{e}'_3. \overline{e}_1 & \overline{e}'_3. \overline{e}_2 & \overline{e}'_3. \overline{e}_3 \end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3} \end{pmatrix} $$

Where the [itex]\overline{e}'_i[/itex]s and [itex]\overline{e}_j[/itex]s are unit basis vectors in the primed and unprimed co ordinate systems respectively.

Now I tried to apply the above idea to the following situation (Riley, Hobson and Bence Chapter 26, Problem 2).

I took [itex]\mathbf{A} = \overline{e}_1[/itex] and [itex]\mathbf{B} = \overline{e}_2[/itex]. It turns out that the matrix that transforms [itex]\mathbf{A} \rightarrow \mathbf{A}' [/itex] and [itex]\mathbf{B} \rightarrow \mathbf{B}' [/itex] is not the matrix that transforms the unprimed components to the primed components (that I used above) but the INVERSE (or transpose) of that matrix. I need to know where I am going wrong here. Thank you.