MATH 218 Final, Fall 2007

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2007-2008 Name:........................................ ID:............................................. Section: 4 (@ 12:30) 5 (@ 9:30) Time: 120 min I- Given the vector space R4 with standard addition and scalar multiplication. Let W = {x ∈ R4 such that Ax = 0} where A =   3 6 0 0 3 7 0 −15 2 4 0 0  . (a) (i) (4 points) Show that W is a subspace of R4. 1 (ii) (5 points) Find a basis for W. (iii) (1 point) Determine the dimension of W. (b) (6 points) Find a basis for W ⊥, the orthogonal complement of W, with respect to the Euclidean inner product. 2 II- Given A =   1 0 0 0 2 5 6 7 3 0 8 9 4 0 0 10   (a) (3 points) Find the eigenvalues of A. (b) (8 points) Find bases for each of the eigenspaces of A. 3 Continue ... 4 (c) (2 points) Explain, without any calculation, whether the column vectors of A are a basis for R4. (d) (1 point) Give without justification a matrix P such that P −1AP is a diagonal matrix. (That is P diagonalizes A) (e) (2 points) If A is the standard matrix representing a linear transformation T and we take M44 with the inner product < A, B >= tr(ATB). Is the function f defined by f(x) = ∥A∥ T(x) a linear transformation? Justify. 5 III- (5 points) Choose one of the following statements and prove it. • Given an m × n matrix A, if Ax = b is consistent for all b then the column vectors of A span Rm. • If a set of nonzero vectors in an inner product space V is orthogonal then it is linearly independent. 6 IV- (a) (8 points) Find a least squares solution for the system:        x −z = 6 2x +y −2z = 27 3x +3y = 0 x +y −z = 3 (b) (4 points) Use part (a) to find ProjW b where b =   6 27 0 3   and W = Span{   1 2 1 1   ,   0 1 1 1   ,   −1 −2 0 −1  } 7 V- (3 points for each part) Give a short answer. 1- The nullity of A =   3 0 1 2 5 0 1 5 2 −1 3 0 5 −6 4   is: 2- The vector space of all diagonal n × n matrices has dimension: 3- If T1 : R2 −→ R2 and T2 : R2 −→ R2 where T1(e1) = (1, 2) T1(2e2) = (0, 6) T2(e1) = T1(e1 + e2) T2(e2) = −3T1(2e1) then T1 ◦ T2(2, −1) is: 4- If T : R2 −→ R2 is the projection on the x-axis then the standard matrix representing T is: 8 5- An orthogonal basis for the space spanned by u1 = (1, 0, 3, 2) and u2 = (−2, 1, 0, 3) with respect to the Euclidean inner product is: VI- (3 points for each part) Use the given to detect the unknown. Justify. 1- A is a matrix where the number of columns is twice the number of rows. The system Ax = b is consistent for all b. The column vectors of A span a subspace of R6. What is rank(A)? 2- A is an n × n matrix with row vectors r1, r2, ..., rn and column vectors c1, c2, ..., cn. The set {r1, r2, ..., rn, c1, c2, ..., cn} is linearly independent. What is ColumnSpaceA? 9 3- S is a set of matrices all of which represent linear transformations from R3 to R3. All linearly independent subsets of S with 8 elements will not span S. What is S? VII- (2 points for each part) Answer TRUE or FALSE. DO NOT JUSTIFY YOUR ANSWER. Ambiguous answers will not be counted. 1- The span of a set of 6 nonzero vectors in any vector space V has infinitely many elements. 2- T : R3 −→ R2 defined by T(x, y, z) = (3x + y, z − 3) is a linear transformation. 3- If V is any inner product space and u and v are any two orthogonal elements of V then ∥u∥2 + ∥v∥2 = ∥u + v∥2. 4- The function f(x) = mx + b is a linear transformation from R to R for any two scalars m and b. 5- Given any inner product space V and any two elements u and v of V we have −∥u∥∥v∥ ⩽ < u, v > ⩽ ∥u∥∥v∥. 6- If T : Rn −→ Rm is a linear transformation and m = n then T is one-to-one. 10 7- If k is a scalar and u is an element of a vector space V where ku = 0 then either k = 0 or u = 0, BUT we can have two nonzero matrices A and B where AB = 0. 8- �5 4 5 7 −5 9 � is a linear combination of �1 2 3 4 0 6 � , �2 −1 0 0 5 −3 � , and �6 1 2 3 0 0 � 9- The set of all 2 × 2 matrices A with det(A) = 0 is a subspace of M22. 10- If S1 and S2 are two subsets of a vector space V and Span(S1) = Span(S2) then S1 = S2. 11- Any subset of a linearly independent set is linearly independent. 12- If {v1, v2, v3, v4} is a basis for a vector space V , then {3v1, 5v2, −v3, −v4} is also a basis for V . 13- An upper triangular matrix of size 5 × 5 has exactly 5 distinct eigenvalues. 14- For any m × n matrix A, the associated normal system is either ATAx = ATb or AATx = ATb since ATA is always symmetric for any A. GOOD LUCK 11