We choose an inner product <|> on the plane such that the green basis vectors are orthonormal. Let T be the linear operator that maps the red vectors to the green ones. The adjoint operator T*: maps every vector p to the vector T*(p) whose inner product with the red vectors is the same as the inner product of p with the the green vectors: . The matrices of T* and T in the green basis are transpose to each other. We denote the green coordinates by .

1. Drag the red points around and observe how the purple vectors change. 2. Find the inner product of each purple vector and each red vector. Conclude that purple vectors are orthogonal to the corresponding red grid lines. Hint: <T*v|u>=<v|Tu>