To every eigenvaluebof a sturm liouville system there corresponds only one linearly independent eigenfunction.

(b) Let C³ - - 1²: B = 1 c² C-3 0 0 с -C² ³+1 " where ce R. How many eigenvalues does B have and with what ge- ometric multiplicities? Without knowing c, is it possible to determine whether B is diagonalizable? Explain your answers.

Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.

9. Determine which of the following vectors 1 -1 is an eigenvector of the matrix 1 1 1 1 -1 1 -1 W = -1 -1 1 -1 -1 1 and find the corresponding eigenvalue.