MATH 218 Final Collection Finals with Solution, Spring 2017

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2005-2006 Name:.................................. ID:................................ Section 1 Section 2 Section 3 Mrs. Z. Mouzeihem Ms. M. Houri Ms. D. Audi Time: 120 min I- (35 points) Let A =    1 2 1 1 3 7 2 2 2 4 2 2    (a) Find a basis for the row space of A. (b) Find a basis for the column space of A. 1 (c) Find a basis for the null space of A. (d) Find a basis for the null space of AT. (e) If W ⊥ is the column space of A deduce a basis for W. 2 II- (25 points) Let T : P2 → P2 be a linear transformation such that: T(p(x)) = p(2x + 3) a) Find the matrix A = [T]B with respect to the standard basis B = {1, x, x2}. b) Find the eigenvalues for the matrix A = [T]B c) Write the formula for [T(p(x))]B without justification. 3 d) Deduce the values of λ such that T(p(x)) = λp(x) has nonzero solutions. III- (20 points) Let S = {u1, u2, u3} be a basis for R3, with u1 = (2, 2, 0), u2 = (2, 0, 0) and u3 = (0, 0, 3) a) Find an orthonormal basis S1 = {q1, q2, q3} for R3. b) Write u = (1, 1, 1) in terms of the basis S1. 4 IV- (20 points) Let T : R3 → R4 be defined by: T(x, y, z) = (x + 2y, x + 3z, z − x, 0) a) Show that T is a linear transformation. b) Is T one-to-one? Justify c) Find the rank of T. d) Find a basis for the range of T. 5 V- (25 points) Let A be a 3 × 3 matrix with eigenvalues λ1 = −2, λ2 = 0 and λ3 = 4, and let v1, v2 and v3 be the corresponding eigenvectors respectively, where v1 = (0, 3, −2), v2 = (−3, −3, 0) and v3 = (−2, 0, 3). Let E1 = span{v1} be the eigenspace corresponding to λ1 = −2, E2 = span{v2} the eigenspace corresponding to λ2 = 0 and E3 = span{v3} the eigenspace corresponding to λ3 = 4. a) Write x = (−10, −3, 4) as a linear combination of the vectors {v1, v2, v3}, and deduce Ax. b) Find the null space of A (Hint: Use the fact that λ = 0 is an eigenvalue for A.) c) Find a matrix P that diagonalizes A and determine P −1AP 6 VI- (15 points) Short Proofs 1) Show that if A is an orthogonal matrix then det(A) = ±1. 2) State and prove the theorem of Pythagoras for an inner product space V . 3) Let T : V → W be a linear transformation, show that Ker(T) is a subspace of V. 7 VII- (30 points) Show your work. a) Let A, B and C be n × n matrices, if B = A−1and C = BT find C−1. b) Let T : R2 → R2 be a linear operator with T(1, 1) = (2, 1) and T(3, 2) = (0, 3), find T(5, 4). c) Let A =    1 0 3 2 5 2 1 0 8    and C =    2 0 0 0 0 2 0 2 0    . If B is a 3 × 3 matrix and AB = C find det(B). d) Let A = � 4 1 d −4 � , find the value of d that makes A orthogonally diagonalizable. 8 e) Find the null space of a 5 × 4 matrix with linearly independent columns. f) Let {u, v, w} be a basis for a vector space V. Show that {w, u + w, u + v + w} are linearly independent. VII- (30 points) Answer TRUE or FALSE without proof (5 points for each right answer and 2 points penalty for each wrong answer). a- Let A be an n × n matrix, if λ = 0 is an eigenvalue for A then A is invertible. b- Let T : Pn → Pn be a linear transformation, if range of T is Pn then T is one-to one. c- Let A, B and C be n × n matrices, if AB = AC then B = C. d- Let A be an m × n matrix and b be an m × 1 vector, if b is not in the column space of A then AX = b has no solution. e- Let T : V → R be a linear transformation with V = span{u}. If T(u) = 0 then T ≡ 0. f- Cramer’s rule can be used to solve any system of linear equations. GOOD LUCK!! 9 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 American University of Beirut Fall 2007-2008 Elementary Linear Algebra (Math 218) Lecturer: Dr. Wassim Daher Section 3, MWF: 1:00 pm - 2:00 pm Final Exam, January 23 2008 Time: 2 hours The quality of your writing, the clarity and the precision of your reasoning, all together will be taken into account in the correction process. Open book and Calculators are not allowed for this Final Exercise 1.(10 Points) Let L : M22 → M22 be the linear operator defined by L(M) = M T Find the matrix for L with respect to the standard basis for M22 Exercise 2.(10 Points) Find (s, t) so    1 0 0 1 1 1    � s t � is close as possible to    1 1 0    Exercise 3.(10 Points) Find an orthogonal basis for the subspace w + 2x + 3y + 4z = 0 of R4. Exercise 4.(15 Points) Let A =    −5 3 4 −2 2 2 −6 3 5    a) Find the eigenvalues of A? Is A diagonalizable?. b) Find a basis to each eigenspace corresponding to the eigenvalues found in part a) and give its dimension. 1 c) calculate A5. Exercise 5.(15 Points) Consider the two subspaces of R3: E1 = {(x, y, z) ∈ R3, 3x − 2y + 3z = 0}, E2 = {(x, y, z) ∈ R3, x + 3y − z = 0 and 3x + 3y + z = 0} a) Find the bases B1 and B2 for E1 and E2 respectively. b) Prove that R3 = E1 ⊕ E2 i.e. 1) E1 ∩ E2 = {−→0 R3} and 2) R3 = E1 + E2 Exercise 6.(10 Points) Consider the bases for P1, B = {p1 = 6 + 3x, p2 = 10 + 2x} and B′ = {q1 = 2, q2 = 3 + 2x} a) Find the transition matrix from B′ to B b) Find the transition matrix from B to B′ c) Compute the coordinate vector [p]B, where p = −4+ x and then find [p]B′ Exercise 7.(20 Points) Let A =    2 −1 −1 −1 2 −1 −1 −1 2    a) Show that A is diagonalizable b) Find the matrix P that orthogonally diagonalizes A. c) Find an orthogonal basis for R3 Exercise 8.(10 Points) Let E be a finite dimensional vector space and let f be a linear operator from E to E. Show the following statement Ker(f) = Ker(f ◦ f) ⇒ Range(f) = Range(f ◦ f) Good Luck 2