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1 Math 6 Final Exam Spring 6 Your name Directions: For each problem, place the letter choice of our answer in the spaces provided on this page B signing here, I pledge that I have neither given nor received assistance on this exam. Your signature

2 . Given that A B, where A = the dimension of Nul(A) is (a) (b) (c) (d) (e) and B =,. Suppose v, v, v, v, v V and v + v v =. I. v, v, v are linearl dependent. II. v, v, v, v, v are linearl dependent. III. v, v are linearl independent. (a) Onl I is true (b) Onl II is true (c) I and II are true (d) Onl III is true (e) I and III are true. Let U = {[ x (a) Onl U is a subspace of IR (b) Onl V is a subspace of IR } {[ IR x x + =. Let V = (c) Both U and V are subspaces of IR (d) Neither U nor V is a subspace of IR. Let T : IR IR be a linear transformation with T (a) T is - but not onto (b) T is onto but not - (c) T is - and onto (d) T is neither - nor onto (e) There is not enough information to decide IR = x }. =. Suppose A is and T (x) = Ax. Suppose dim(ker(t )) =. I T is not - II. T is not onto (a) Onl I is true (b) Onl II is true (c) I and II are both true (d) There s not enough information to decide. Then

3 [ a 6. Let A = a a. Then A is not invertible if (a) a = (b) a = (c) a = or a = (d) a = or a = 7. Determine the values of h and k so that [ h k is the augmented matrix of a consistent linear sstem with infinite solutions. (a) h = 9, k = 6 (b) h = 9, k 6 (c) h 9, k IR (d) h 9, k 6 8. Suppose V is a vector space with dimension n. I. Ever set of m < n vectors in V is linearl independent. II. Ever set of m > n vectors in V spans V. (a) Onl I is true (b) Onl II is true (c) I and II are both true (d) I and II are both false 9. Suppose x = [ x IR and T ([ x I. T : IR IR II. T : IR IR III. T is a linear transformation IV. T is not a linear transformation (a) I and III are true (b) I and IV are true (c) II and III are true (d) II and IV are true ) = x.. Let S = {p, p, p } be a set of polnomials in IP, let A be the matrix whose columns are the coordinate vectors of the polnomials, and suppose A B, where B =. Then (a) S spans IP, but the polnomials in S are not linearl independent. (b) The polnomials in S are linearl independent, but S does not span IP (c) S is a basis for IP (d) S is an orthogonal basis for IP

4 . Warning: Look at the elements of S ver carefull. Let x S = = s + t z 6 Then S is (a) A plane in IR not through (b) A plane in IR not through (c) A line in IR through (d) A line in IR not through The dimension of the vector space M 7 9 is (a) 7 (b) 9 (c) 6 (d) 6. Let A and B be matrices. Suppose T : IR IR is given b T (x) = Ax and T : IR IR is given b T (x) = Bx. Then the map defined b T (x) = (AB) T (x) is (a) A linear transformation from IR to IR (b) A function from IR to IR but not a linear transformation (c) A linear transformation from IR to IR (d) not defined. Let A =. One eigenvector of A is (a) (b) (c) (d). For the eigenvalue λ = of the matrix A = 6 (a) (b) (c) (d) is not an eigenvalue of A, the eigenspace is k-dimensional, where k =

5 6. For an n n diagonalizable matrix A, identif the statement that is not necessaril true. (a) A has n linearl independent eigenvectors. (b) The multiplicit of each eigenvalue of A is equal to the dimension of the corresponding eigenspace. (c) AP = P D, where P is made of eigenvectors of A and D is a diagonal matrix whose entries correspond to the columns of P. (d) The eigenvectors of A form a basis of IR n. (e) A has n distinct eigenvalues. [ 7. The matrix A = has complex eigenvalues. For an x IR, the computation Ax serves to rotate x b an angle of π counterclockwise, and it scales x b a factor of... (a) (b) (c) (d) The angle of rotation is not π counterclockwise. 8. A is a real matrix with eigenvalues λ =,,,, and k. The eigenspaces for λ = and are each two-dimensional. Then A is guaranteed to be diagonalizable if (a) k (b) k, (c) k,, (d) k is complex 9. Let B be a basis for IR, let D be a basis for IP, and let T : IR IP be a linear transformation. Then the matrix of T relative to B and D is (a) (b) (c) (d). Let T : IR M bea linear transformation, and recall that the standard bases for IR and M are,,, {[ [ [ [ }, and,,,, respectivel. Suppose the matrix of T relative to the standard bases of IR and M is 9 Then (a) 8 (b) 6 78 (c) [ 8 T (d) = [ 6 78

6 . Let T : IP IP be defined b T (p(t)) = p (t) + (t )p (t). Let B = {, t, t, t } be the standard basis of IP and let D = { + t, t, + t } be a basis of IP. The third column of the matrix of T relative to B and D is [ / 6 6 (a) (b) / (c) (d). Let A = and b = 9. A least squares solution to Ax = b is (a) [ 9 (b) (c) [ 6 (d). Let A = (a) (b) and b = (c) 6. Then proj Col(A) b = /7 /7 6/7 (d) [ [. Let A =, and suppose for some b IR, proj 6 Col(A) b = The set of least squares solutions to Ax = b is a (a) point (b) plane not through the origin (c) plane through the origin (d) line not through the origin.. Let A be an m n matrix and let b IR m. I. There will alwas be at least one least squares solution to Ax = b, even if b is not in Col(A). II. A least squares solution ˆx to Ax = b is the closest point to b in Col(A). III. If ˆx is a least squares solution to Ax = b, then Aˆx b is the least squares error. (a) Onl II is true (b) Onl III is true (c) II and III are true (d) I and III are true (e) I, II, and III are all true 6