We choose an inner product <|> on the plane such that the green basis vectors are orthonormal.[br]Let T be the linear operator that maps the red vectors to the green ones. [br]The adjoint operator T*:[math]\mathbb{R}^2\mapsto\mathbb{R}^2[/math] maps every vector [b]p[/b] to the vector T*([b]p[/b]) whose inner product with the red vectors is the same as the inner product of [b]p[/b] with the the green vectors: [br] [math]<{T^*\bf p}|{\bf u}_i>=<{\bf p}|{T\bf u}_i>=<{\bf p}|{\bf v}_i>[/math].[br]The matrices of T* and T in the green basis are transpose to each other. [br][br]We denote the green coordinates by [math]x_1,x_2[/math].

1. Drag the red points around and observe how the purple vectors change.[br]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.[br]Hint: <T*[b]v[/b]|[b]u[/b]>=<[b]v[/b]|T[b]u[/b]>