Exam Sitting For Linear Algebra (Mast10007)

P ag e 1 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 1 of 25 Student number Semester 2 Assessment, 2021 School of Mathematics and Statistics MAST10007 Linear Algebra Reading time: 30 minutes — Writing time: 3 hours — Upload time: 30 minutes This exam consists of 25 pages (including this page) with 12 questions and 125 total marks Permitted Materials • This exam and/or an offline electronic PDF reader, one or more copies of the masked exam template made available earlier and blank loose-leaf paper. • One double sided A4 page of notes (handwritten only). • No calculators are permitted. No headphones or earphones are permitted. Instructions to Students • Wave your hand right in front of your webcam if you wish to communicate with the supervisor at any time (before, during or after the exam). • You must not be out of webcam view at any time without supervisor permission. • You must not write your answers on an iPad or other electronic device. • Off-line PDF readers (i) must have the screen visible in Zoom; (ii) must only be used to read exam questions (do not access other software or files); (iii) must be set in flight mode or have both internet and Bluetooth disabled as soon as the exam paper is downloaded. Writing • Marks are awarded for – Using appropriate mathematical techniques. – Showing full working, including results used. – Accuracy of the solution. – Using correct mathematical notation. • If you are writing answers on the exam or masked exam and need more space, use blank paper. Note this in the answer box, so the marker knows. • If you are only writing on blank A4 paper, the first page must contain only your student number, subject code and subject name. Write on one side of each sheet only. Start each question on a new page and include the question number at the top of each page. Scanning and Submitting • You must not leave Zoom supervision to scan your exam. Put the pages in number order and the correct way up. Add any extra pages to the end. Use a scanning app to scan all pages to PDF. Scan directly from above. Crop pages to A4. • Submit your scanned exam as a single PDF file and carefully review the submission in Gradescope. Scan again and resubmit if necessary. Do not leave Zoom supervision until you have confirmed orally with the supervisor that you have received the Gradescope confirmation email. • You must not submit or resubmit after having left Zoom supervision. c©University of Melbourne 2021 Page 1 of 25 pages Can be placed in Baillieu Library P ag e 2 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 2 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 1 (11 marks) Consider the linear system x+ 2y + 3z = 6 4x+ 5y + 6z = 15 7x+ 8y + 9z = 24 (a) Reduce the augmented matrix [A|b] of the system to reduced row-echelon form. Indicate the row operations used in each step. (b) Determine the ranks of A and [A|b] and interpret your answer. (c) Find the solution set of the system. (d) Find the solution set of the corresponding homogeneous system. (e) Prove that the set of all choices for c such that [A|c] can be solved, forms a plane through the origin in R3. (f) Find a normal vector to the plane in part (e). Page 2 of 25 pages P ag e 3 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 3 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 3 of 25 pages P ag e 4 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 4 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 2 (8 marks) (a) Let A be an n× n matrix. Prove that the determinant of A equals zero if and only if the nullity of A is greater than or equal to 1. (b) Let u ∈ R3, and consider the linear transformation T : R3 → R3 such that T (v) = u× v. What is the determinant of [T ]? Justify your answer. Page 4 of 25 pages P ag e 5 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 5 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 5 of 25 pages P ag e 6 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 6 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 3 (14 marks) Let L1 be the line x− 3 −1 = y − 8 −2 = z + 2 and let L2 be the line x = 4− 3s, y = −1 + 5s, z = −4 + 4s, s ∈ R. (a) Find the point of intersection of L1 and L2. (b) Find the angle between the lines L1 and L2. (c) Find a Cartesian equation of the plane in R3 that contains both L1 and L2. Page 6 of 25 pages P ag e 7 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 7 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 7 of 25 pages P ag e 8 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 8 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 4 (9 marks) Let V = R2 be the real vector space with vector addition ⊕ defined by (x1, x2)⊕ (y1, y2) = (x1 + y1 − 1, x2 + y2 − 2) for each (x1, x2), (y1, y2) ∈ V and scalar multiplication � defined by α� (x1, x2) = (αx1 − α+ 1, αx2 − 2α+ 2) for each (x1, x2) ∈ V and α ∈ R. (a) Verify the associative property u⊕ (v ⊕w) = (u⊕ v)⊕w when u = (−3, 1), v = (5, 4), w = (2, 7). (b) Prove 1� u = u for all u ∈ V . (c) Prove the distributive property (α+β)�u = (α�u)⊕ (β�u) for all u ∈ V and α, β ∈ R. (d) Find the zero vector 0 in V . Show that u⊕ 0 = u for all u ∈ V . Page 8 of 25 pages P ag e 9 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 9 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 9 of 25 pages P ag e 10 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 10 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 5 (12 marks) Determine if the following sets are subspaces of the real vector space M2,2 of 2× 2 matrices. If the set is a subspace, you must use the subspace theorem to prove it. If the set is not a subspace, then you must provide an explicit counter-example. (a) The set S of all matrices A such that AD = DA, where D = [ 1 2 3 4 ] . (b) The set M of all matrices of the form [ s+ 3 s+ t 0 2t+ 6 ] where s, t ∈ R. (c) The set W of all matrices of the form [ r 0 0 r2 ] where r ∈ R. Page 10 of 25 pages P ag e 11 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 11 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 11 of 25 pages P ag e 12 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 12 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 6 (12 marks) Let W be the subspace of P3 spanned by S = {1 + 2x+ 3x3, 2 + 4x+ 6x3, 1 + x2 + x3, 5 + 3x+ 5x2 + 8x3, 3− 5x+ x2 − 2x3}. You may assume that A =  1 2 1 5 3 2 4 0 3 −5 0 0 1 5 1 3 6 1 8 −2  ∼  1 2 0 0 2 0 0 1 5 1 0 0 0 3 −9 0 0 0 0 0  = B. (a) Is W equal to P3? Explain. (b) Find a basis B for W consisting of polynomials in S. (c) What is the dimension of W? (d) For each vector in S that is not in B, find the coordinate vector with respect to the basis B. (e) Let C = {2 + 4x+ 6x3, 1 + x2 + x3, 3− 5x+ x2 − 2x3}. (i) Is C linearly independent? Explain. (ii) Is C a basis for W? Explain. Page 12 of 25 pages P ag e 13 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 13 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 13 of 25 pages P ag e 14 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 14 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 7 (10 marks) Let P2 be the vector space of real polynomials of degree ≤ 2 with ordered basis B = {1, x, x2}. Let T : P2 → R4 be the linear transformation defined by T (p(x)) = (p(0),p(1),p(−1),p(3)). (a) What is T (x2 − 3x)? (b) Find the matrix representation of T with respect to the basis B and the standard basis S of R4. (c) Compute the rank of T . (d) Is T surjective? Explain your answer. (e) Is T injective? Explain your answer. Page 14 of 25 pages P ag e 15 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 15 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 15 of 25 pages P ag e 16 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 16 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 8 (9 marks) For each of the following linear transformations, determine the (i) image (ii) kernel (iii) inverse transformation, if it exists. (a) Reflection R : R2 → R2 in a line ` ⊂ R2 passing through the origin. (b) The map P : R3 → R3 given by orthogonal projection onto a plane Π through the origin in R3. (c) The differentiation function D : P3(R)→ P2(R) mapping a polynomial p(x) to its deriva- tive dp dx . Hint: You do not need to find the matrix representations of the transformations. Page 16 of 25 pages P ag e 17 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 17 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 17 of 25 pages P ag e 18 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 18 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 9 (14 marks) Consider the complex matrix A = [ 6 2 + 2i 2− 2i 4 ] . (a) Show that A is a Hermitian matrix. (b) Find the eigenvalues of A. (c) Find the eigenspace corresponding to each eigenvalue of A. (d) Find a unitary matrix U and a diagonal matrix D such that A = UDU−1. Page 18 of 25 pages P ag e 19 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 19 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 19 of 25 pages P ag e 20 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 20 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 10 (8 marks) (a) Let A be an arbitrary 2× 2 matrix. (i) Determine the characteristic equation of A. (ii) Show that the characteristic equation of A can be written as λ2 − Tr(A)λ+ det(A) = 0. (b) Let V be a real vector space equipped with an inner product 〈u,v〉. Using the inner product space axioms, prove that for any u,v ∈ V, ‖u + v‖2 + ‖u− v‖2 = 2‖u‖2 + 2‖v‖2. Page 20 of 25 pages P ag e 21 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 21 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 21 of 25 pages P ag e 22 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 22 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 11 (9 marks) Let u = (u1, u2) ∈ R2, v = (v1, v2) ∈ R2 and 〈u,v〉 = 3u1v1 − u1v2 − u2v1 + 6u2v2. (a) Write 〈u,v〉 in terms of a symmetric matrix. (b) Show that 〈u,v〉 defines an inner product on R2. (c) Find the angle between u = (2, 1) and v = (1, 2) with respect to this inner product. Page 22 of 25 pages P ag e 23 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 23 of 25 MAST10007 Linear Algebra Semester 2, 2021 Page 23 of 25 pages P ag e 24 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 24 of 25 MAST10007 Linear Algebra Semester 2, 2021 Question 12 (9 marks) Find an orthonormal basis for P2 with respect to the inner product 〈p,q〉 = ∫ 1 −1 p(x)q(x) dx by applying the Gram Schmidt algorithm to the basis {v1,v2,v3} = {1, x, x2}. Page 24 of 25 pages P ag e 25 of 25 — ad d an y ex tr a p ag es af te r p ag e 25 — P ag e 25 of 25 MAST10007 Linear Algebra Semester 2, 2021 End of Exam — Total Available Marks = 125 Page 25 of 25 pages