Final A Math115A Nadja Hempel 03/23/2017 nadja@math.ucla.edu Name: UID: Problem Points Score 1 10 2 20 3 5 4 5 5 9 6 5 7 7 8 13 9 16 10 10 Total 100 1
2 Exercise 1. (10pt) Let T : V V be a linear transformation. (1) Define λ is an eigenvalue for T with corresponding eigenvector v. (2) Define the characteristic polynomial of T. (3) Define the eigenspace E λ of an eigenvalue λ for T. (4) Define the algebraic multiplicity of an eigenvalue λ for T. (5) Define: T is diagonalizable.
3 Exercise 2. (20pt) Let T : M 2 2 (R) M 2 2 (R) defined by Let B be the standard basis of M 2 2 (R), i. e. and let B = B = ( ) ( ) a b 2a + b c. c d b 2d + c {( ) 1 0, 0 0 {( ) 1 1, 0 0 ( ) 0 1, 0 0 ( ) 1 2, 0 0 (1) Show that B is a basis for M 2 2 (R). (2pt) ( ) 0 0, 1 0 ( ) 1 0, 1 0 ( )} 0 0 0 1 ( )} 0 0 0 1 (2) Compute [T ] B and [T ] B. (2pt + 4pt)
4 (3) Let A = ( ) 1 1. Compute [T (A)] 1 1 B. (3pt)
5 (4) Compute the eigenvalues of T and the corresponding eigenspaces. (6pt) (5) Is T diagonalizable? Justify your answer. (3pt)
6 Exercise 3. (5pt) Let T : P 3 (R) P 3 (R) be given by T (f) = (x + 2)f (x). (1) Prove that T is a linear transformation. (2pt) (2) Is T is one-to-one? Justify your answer. (1pt + 2pt)
Exercise 4. (4pt) Let A and B be n n matrices such that AB = BA. Fix a non-zero eigenvalue λ of A with eigenvector v. Prove the following: (1) Bv is an eigenvector of A with eigenvalue λ. (2pt) 7 (2) Conclude that if v is not an eigenvector of B, then the algebraic multiplicity of λ for A is bigger than 1. (3pt)
8 Exercise 5. (9pt) Let V be vector space and T be a linear operator on V. Recall that T can be restricted to a subspace W, if T (W ) W. We denote this restriction by T W : W W Let {λ 1,..., λ n } be the set of eigenvalues of T and let W = E λ1 + + E λn. Let a i be the algebraic multiplicity of λ i and m be the sum of the the a i s. (1) Show that W = E λ1 E λn. (2pt) (2) Show that for any vector v W, we have that T (v) W. (i. e. T is W -invariant) (2pt)
9 (3) Conclude that T W is diagonalizable. (1pt) (4) Suppose that Y is another T -invariant subspace of V such that T Y is diagonalizable. Show that dim(y ) m. (4pt)
10 Exercise 6. (5pt) Let W 1 and W 2 be two subspaces of V such that dim(w 1 ) dim(w 2 ). Show that W 1 W 2 is a subspace of V if and only if W 1 is contained in W 2.
11 Exercise 7. (7pt) Consider the vector space P 1 (R) with the inner product given by f, g = Do the following: (1) Compute x. (2pt) 2 0 f(x)g(x) dx (2) Compute the orthogonal projection of x onto the subspace generated by 1. (3pt) (3) Use the previous part to find an orthonormal basis for P 1 (R) with the inner product defined above. (2pt)
12 Exercise 8. (13 pt) Let V be an inner product space and T be a linear operator on V. Let λ be an eigenvalue of T, v be a corresponding eigenvector to λ and W = span{v}. (1) Show that v is in R(T λi). (2pt) (2) Conclude that T λi is not onto. (1pt) (3) Conclude that λ is an eigenvalue for T and that T has an eigenvector. (2pt)
(4) Show that W as well as W are T -invariant, i. e. T (W ) W and T (W ) W. (2pt) 13 Now suppose that the characteristic polynomial of T splits. Show that (5) The characteristic polynomial of T W : W W divides the one of T. (3pt)
14 (6) Using all of the above, show by induction on dim(v ) that V has an orthonormal basis B such that [T ] B is an upper triangle matrix. (5pt)
15 Exercise 9. (16pt) Answer the following questions and justify your answer. For what values of a, b, c, d is: {(x, y, z) : x + y = a, bxy = 0} a subspace of R 3. (2pt) {f(x) P (R) : f (x) = a, f(x) = bx 4 + c(x 3 + x 2 ) + d} a subspace of P 3 (R). (2pt) T : R 3 R 3, define by (x, y, z) (ax + y, by 2, z). (2pt) U : R 3 P 2 (R), defined by (x, y, z) axt 3 + byt 2 + czt + d. (2pt)
16 {(a, b, 0), (a, c, 0), (1, 1, d)} a basis for R 3. (2pt) T : R 2 R 3 defined by (x, y) (ax + by, c(x + y), x + y) not injective? (3pt) U : R 3 R 2 defined by (x, y, z) (ax + by + cz, x + z) not surjective? (3pt)
17 Exercise 10. (10pt) Circle T or F to mark the following statements as true or false. T / F : If v is an eigenvector of a transformation T with eigenvalue 12, then 3v is an eigenvector with eigenvalue 36. T / F : A matrix A is invertible if and only if det(a) = 0. T / F : Let T : V V be a linear transformation. If for every vector eigenvalue λ the dimension of the eigenspace E λ is equal to the algebraic multiplicity of λ, then T is diagonalizable. T / F : Let T : V W be a linear transformation and let B be a basis for V. R(T ) = span({t (v) v B}). Then T / F : Let S : U V and T : V W be linear transformations. Let A, B and C be ordered bases for U, V and W respectively. The following is true: [T S] C A = [T ]C B [S]B A. T / F : Let A be an n n matrix. If A 2 = 0 then A = 0 where 0 is the n n matrix of all zeros. T / F : If for two matrices A and B we have that AB = I, then A and B are invertible. T / F : The determinant of an upper triangular matrix is the product of its diagonal entries. T / F : Let A be an n n matrix. If A is diagonalizable, then A is invertible. T / F : Let V be an inner product space with basis B = {v 1,..., v n }. Then (1, 0,..., 0) = 1. (Note: (1, 0,..., 0) is the coordinate vector in the basis B)