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 Incoming links: Reciprocal-basis coordinate transformation

u_{i}

a = \sum_{j} a_{j}^{'} w_{j}

. The reciprocal basis condition gives

a_{i}^{'} = ⟨ u_{i}, a ⟩

. Expanding instead in the

u

a_{i}^{'} = j \sum G_{ij} a_{j}, G_{ij} = ⟨ u_{i}, u_{j} ⟩ .

Thus

G

In the specified two-dimensional Cartesian frame,

u_{1} = (1, 0), u_{2} = (\frac{1}{2}, \frac{3}{2}) .

Solving the four dot product conditions gives

w_{1} = (1, - \frac{1}{3}), w_{2} = (0, \frac{2}{3}) .

u_{2}

and

u_{1}

, respectively, with lengths adjusted so the other dot product is one. Equating the Cartesian components of

a = a_{1} u_{1} + a_{2} u_{2} = a_{1}^{'} w_{1} + a_{2}^{'} w_{2}

gives

a_{x} = a_{1} + \frac{1}{2} a_{2} = a_{1}^{'}, a_{y} = \frac{3}{2} a_{2} = \frac{- a _{1}^{'} + 2 a _{2}^{'}}{3} .

Consequently

(a_{1}^{'} a_{2}^{'}) = (1 1/2 1/2 1) (a_{1} a_{2}),

which agrees entry by entry with the Gram matrix computed directly from the two unit basis vectors.