[b]Active viewpoint (upper picture)[/b]: T is a linear transformation of the plane that maps the red basis vectors to the green ones. Its matrix A=[T] is the same in either basis.[br][b][br]Passive viewpoint (lower picture)[/b]: Every vector [math]\vec{v}[/math] has two sets of coordinates: relative to the green basis [math]\vec{v}=x_1\vec{v}_1+x_2\vec{v}_2[/math] and relative to the red basis [math]\vec{v}=y_1\vec{u}_1+y_2\vec{u}_2[/math]. Matrix A describes the change of coordinates from green to red. [br]We choose an inner product such that the green vectors are orthonormal. If T* is the adjoint transformation to T with respect to this inner product, then the matrix of T* in the green basis is transpose of A. [br]See the attached PDF file for details. [br]

1. Drag the green points around and observe how the purple vectors move.[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: Observe that [math]\vec{u}_i=T^{-1}\vec{v}_i[/math] and <T*[b]v[/b]|[b]u[/b]>=<[b]v[/b]|T[b]u[/b]> ...