1) Find the transition matrix PST and the coordinates [v]s, where T is the natural basis {e1, e2, e3, e4} in R4 and -2 1 0 1 1 -2 1 S = -2 1 0 1 0 -2 3 Solution: By the definition, -2 1 0 0 1 -2 1 1 Pr+s= 0 1 -2 0 0 1 0 -2 is the matrix composed by the given vectors of the S-basis. Hence, PS+T=PT+S = -1 -2 -1 1212 - -1 - 21121 (I assume that you remember how to find the inverse matrix; as a self-checking hint, the inverse of a symmetric matrix must be symmetric) and [V]S = PS+T[V]T = -2 -3 -2 -3 2323