Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your notes or text for this exam, but you may use a calculator. Good Luck! Problem Points Score 4 3 9 4 8 5 8 6 7 8 9 5
() Let A = For what follows, you can use a calculator to check your work, but you must 3 show work justifying your answer to receive credit. (a) Find the eigenvalues of A. Solution: We have ( λ) det(a λi) = det ( λ) = (3 λ) det (3 λ) = (3 λ)(λ 4λ + 3) = (3 λ)(λ 3)(λ ) ( λ) ( λ) [3 pts] We see that the eigenvalues of A are: λ = 3 (algebraic multiplicity ) and λ = (algebraic multiplicity ). (b) For each eigenvalue of A, find the corresponding eigenspace. [6 pts] Solution: To find E 3 we solve for null(a 3I). We row reduce the associated augmented matrix: [A 3I ] = Solving the system, we see that E 3 = Span,. To find E we solve for null(a I). We row-reduce the associated augmented matrix: [A I ] = Solving the system, we see that E = Span. (c) Is A diagonalizable? D = P AP. If not, explain why it is not. If so, find an invertible matrix P and a diagonal matrix D such that [5 pts] Solution: We see that the algebraic multiplicity equals the geometric multiplicity for each eigenvalue of A. Thus, A is diagonalizable. We let P = and D = 3 3
() Let A = 3 4. 3 3 6 For what follows, you can use a calculator to check your work, but you must show work justifying your answer to receive credit. (a) Without row-reducing, use Laplace expansion to find the determinant of A. [8 pts] Solution: We begin with Laplace expansion down the third column of A: 4 det(a) = 3 det 3 3 6 = 3 ( ) det 3 + (3) det 4 6 3 = 3[( )( 8) + 3(6 8)] = 3(3 6) = 78 (b) det(a ) = det(a) = 78 [ pt] (c) det(a T ) = det(a) = 78 [ pt] (d) det(a ) = (det(a)) = ( 78) = 684 [ pt] (e) det(a) = 4 det(a) = 6( 78) = 48 [ pt] 3
(3) Consider the inconsistent system x y = x + y = 4 x 4y = (a) What are the normal equations for the least squares approximating solution? Solution: Let A = and b = 4 Then 4 A T A = 4 4 = 6 6 [3 pts] and A T b = 4 4 = 7 So, the normal equations are A T Ax = A T b or 6 x = 6 7 (b) Find the least squares approximating solution. [6 pts] Solution: We need to solve the system given by the normal equations in part (a). We row-reduce the associated augmented matrix: 6 6 6 7 8 Solving the system, we see that x = 7/6 /3 4
8 8 38 (4) Let A = Fact: A row reduces to 3 Use the Gram-Schmidt Process to find an orthogonal basis for the column space of A. [8 pts] Solution: The first three columns, call these vectors x, x, x 3 respectively, of A form a basis for col(a). We apply the Gram-Schmidt Process to these to obtain an orthogonal basis {v, v, v 3 } for col(a). We start by letting v = x = Finally, let Now let v = x x v v = v v v 3 = x 3 x 3 v v x 3 v v = v v v v (5) Let W be a subspace of R n. 8 8 36 9 36 9 = 8 4 8 8 (a) Complete the definition: The orthogonal complement of W is the set 8 4 = [ pts] Solution: W := {v in R n : v w = for all w in W }. (b) Find a basis for W if W = span,,. [6 pts] 3 Solution: W = null(a T ) where A is the matrix whose columns are the vectors in the spanning set for W. We row-reduce the augmented matrix: [A T ] = 3 Solving the system, we let x 4 = t and x = s. Then x 3 = x 4 = t and x = x + x 3 x 4 = s t. So, a basis for W = null(a T ) is,. 5
(6) For each of the following, determine whether W is a subspace of M (using the usual definitions of matrix addition and scalar multiplication). If W is a subspace of M, then find dim(w ). If W is not a subspace of M, give an explicit example showing how it fails to be one. [6 pts each] (a) W = {A M : rank(a) = } Solution: W is not a subspace of M. To see this, note that A = and B = are both in W, but A + B = is not in W since rank(a + B) =. (b) W = a b : a + c = b + d c d Solution: W is a subspace of M. To see this, let A = a b and B = e f be in W c d g h and α be a scalar. Then (a + e) (b + f) A + B = (c + g) (d + h) and (a + e) + (c + g) = (a + c) + (e + g) = (b + d) + (f + h) = (b + f) + (d + h). We conclude that A + B is in W. Also, αa = αa αc and αa + αc = α(a + c) = α(b + d) = αb + αd. This shows that αa is also in W. To find dim(w ), we need to find a basis for W. Notice that W = b + d c b c d = b + d + c = Span,, αb αd By inspection, if c, c and c 3 are scalars such that c + c + c 3 = then c = c = c 3 =. Thus, B =,, Therefore, dim(w ) = 3. is a basis for W. 6
(7) The sets B = {, + x, ( + x) } and C = { + x, x + x, + x } are two bases for P. (a) Find the change of basis matrix P C B. Be sure to show all of your work. [9 pts] Solution: By inspection, we see that = ( + x) (x + x ) + ( + x ) + x = ( + x) + (x + x ) + ( + x ) + x + x = ( + x) + ( + x ) + ( + x ) So, by definition, P C B = / / / (b) Use your answer to part (a) to express p(x) = 7 + 5x + 4x as a linear combination of the polynomials in C. [3 pts] Solution: By inspection, we see that p(x) = 7 + 5x + 4x = 4( + x + x ) 3( + x) + 6(). So, Thus, / 6 [p(x)] C = P C B [p(x)] B = / 3 / 4 p(x) = 4( + x) + ( + x ) + 3( + x ). = 4 3 7
(8) Let U, V, W be vector spaces. Let T : U V and S : V W be transformations. (a) Complete the definition: T : U V is a linear transformation if [ pts] Solution: T (x + y) = T (x) + T (y) and T (cx) = ct (x) for all x, y in V and all scalars c. (b) Suppose that S and T are both linear transformations. Prove that S T : U W is a linear transformation. [8 pts] Solution: Let x, y be two vectors in U and let α be a scalar. Then (S T )(x + y) = S(T (x + y)) = S(T (x) + T (y)) = S(T (x)) + S(T (y)) = (S T )(x) + (S T )(y) and (S T )(αx) = S(T (αx)) = S(αT (x)) = αs(t (x)) = α(s T )(x) 8
(9) Are the following statements true or false? Carefully justify your answers. [3 pts each] (a) If the 4 4 matrix A has eigenvalues λ = and λ = each having algebraic and geometric multiplicity, then A 5 = I 4. Solution: This statement is true. The matrix A must be diagonalizable. Let P be the matrix of linearly independent eigenvectors for A and D be the corresponding diagonal matrix whose diagonal entries are the eigenvalues of A. Then D 5 = I 4 and A = P DP = A 5 = P D 5 P = P I 4 P = I 4. (b) If the n n invertible matrix A has eigenvalue λ = 3, then A has eigenvalue 3. Solution: This statement is false. For example, let A = 3. Then A = /3. We see that is an eigenvalue of A but not of A. (c) There exists a subspace W of P such that dim(w ) = 4. Solution: This statement is false. Any subspace W of P must satisfy dim(w ) dim(p ) = 3. (d) Suppose T : V V is a linear transformation and let {v, v,..., v k } be a basis for V. If T (v ) = v, T (v ) = v,..., T (v k ) = v k, then T is the identity transformation on V (i.e., T (x) = x for all x in V ). Solution: This statement is true. Let x be n V. Then we can find scalars c,..., c k such that x = c v + c v + + c k v k. Thus, T (x) = T (c v + + c k v k ) = c T (v ) + + c k T (v k ) = c v + + c k v k = x. (e) If A is an orthogonal matrix, then det(a ) = det(a). Solution: This statement is true. If A is orthogonal, then A = A T. So, det(a ) = det(a T ) = det(a). 9