I Expanding a given vector into another orthonormal basis

[Pushoam]

Equation 9.2.25 defines the inner product of two vectors in terms of their components in the same basis.
In equation 9.2.32, the basis of $|V \rangle$ is not given.
$|1 \rangle$ and $|2 \rangle$ themselves form basis vectors. Then how can one calculate $\langle 1| V \rangle$ ?
Do we have to assume that all of these vectors are given in the orthogonal basis with vectors given by
$\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ ?

 

[jasonRF]

The author is assuming that the components of all of the vectors in 9.2.32 and 9.2.33 are with respect to the same basis. They are being somewhat sloppy in their writing, which is probably the source of your confusion.

jason