(Change of basis) Let be an orthonormal basis of Vn, so that (1.114) and (1.115) hold for...

(Change of basis) Let be an orthonormal basis of Vn, so that (1.114) and (1.115) hold for an arbitrary vector v. In particular, we can write

a) Show that, with A = (aij),

Thus the matrix A provides the link between the components of v with respect to the

two orthonormal bases. [Hint: Write and dot both sides with ei* for I = 1, …, n.]

b) Show that the j th column of A gives the components of ej with respect to the basis and the ith row of A gives the components of ei* with respect to the basis

c) Show that A is an orthogonal matrix (Section 1.13).