Let A be an n x n symmetric matrix. Use Exercise 21 and an eigenvector basis for Rn to give a second proof of the decomposition in Exercise 20(b).

Data From Exercise 21

Show that if v is an eigenvector of an n x n matrix A and v corresponds to a nonzero eigenvalue of A, then v is in Col A.

Data From Exercise 20(b)
b. Show that each y in Rⁿ can be written in the form y = ŷ + z, with ŷ in Col A and z in Nul A.