(Change of coordinates in En) We choose a new origin O* and a new orthonormal basis for V...

(Change of coordinates in En) We choose a new origin O* and a new orthonormal basis for Vn as in Problem 1. Then each point P obtains new coordinates by the equation

a) Let O* have old coordinates Show with the aid of the results of Problem 1 that

Where A is an orthogonal matrix.

b) Show that if the origin is changed but there is no change of basis (so that one has a translation of axes), then

Problem 1

(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).