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Microsoft Word - exam1.sample.(up to 4.4).doc MATH 2362 SAMPLE EXAM 1 NAME: ________________________ STUDENT NUMBER: __________________ 1. There are 7 questions on 7 pages – CHECK YOURS NOW. 2. The time allotted to write the examination is 50 minutes. 3. Answer each question in the space provided – use the back of pages as needed. 4. Give exact answers where possible. 5. For full marks all relevant work should be included. 6. Good luck! 1 6 pts 2 8 pts 3 20 pts 4 4 pts 5 7 pts Total 45 pts MATH 2362 Sample Exam 1 Page 1 of 7 1. Answer the following questions for the system of equations given below. x1 − 5x2 + 6x3 + 2x4 = 22 3x1 − 15x2 + 20x3 +10x4 = 78 2x1 −10x2 + 10x3 = 32 a) Write down the augmented matrixA . [1] b) Write down the reduced row echelon form of your system. [2] c) Find the solution of the system of equations and express the solution in the form p h+x x where x p is a particular solution and xh is the solution of the homogeneous system. [3] MATH 2362 Sample Exam 1 Page 2 of 7 2. A square matrix A is called skew-symmetric if T = −A A . a) Is 0 2 3 2 1 7 3 7 0 − −⎡ ⎤ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥−⎣ ⎦ A skew-symmetric? Explain. [1] b) Assume that A is a skew-symmetric matrix. Is 2=B A skew-symmetric, symmetric or neither? Explain. [3] c) Show that the product of two 2 2× skew-symmetric matrices is symmetric. Is this true in general for all n n× skew-symmetric matrices? Explain. [4] MATH 2362 Sample Exam 1 Page 3 of 7 3. Decide if the following questions are true or false. Circle your answer. Fully explain your answers. [20] a) Given two matrices A and B where AB is defined, if B has a zero column, then AB has a zero column. b) Let A be an n n× matrix, n ≥ 2 , then c c=A A . c) Given a homogeneous system =Ax 0 and its two solutions u and v , then 0.3 0.5+u v is also a solution of the given homogeneous system. d) If an n n× matrix A has no zero rows, then rref A( ) = In . e) Given n n× matrices A and B such that =AB A , then it is necessary that n=B I . f) If A is symmetric and invertible, then 1 T− =A A . g) If the coefficient matrix of a system =Ax 0 is 2n n× then the system has only a trivial solution. MATH 2362 Sample Exam 1 Page 4 of 7 4. Do ONE of the following two proofs. [4] A. Show that 2 21 1|| || || || 4 4 = + − −uv u v u v for any two vectors u,v∈ℜn or B. If A is an n n× matrix then adj A = A n−1 . MATH 2362 Sample Exam 1 Page 5 of 7 5. a) Expand the determinant along the 3rd column. Do NOT compute. [2] 0 −1 3 1 1 7 −2 −2 0 b) Let 2 0 1 3 7 2 1 1 0 −⎡ ⎤ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎣ ⎦ A and 12 22 2 1 7 adj 7 4 2 14 A A − −⎡ ⎤ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥− −⎣ ⎦ A . Find cofactors 12A and 22A . [3] c) Let ( )adj=B A A. Find b33. What conclusion can we make from this result? [2] MATH 2362 Sample Exam 1 Page 6 of 7 6. Define what it means for a subset W of a vector space V to be a subspace. Decide whether each subset is a subspace of the given vector space. a) W = a 0 a2 c ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ : a,c ∈! ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ ⊂ M22 b) W = B ∈M22 : AB = BA{ }⊂ M22 , where A ∈M22 c) W = x, y : y ≤ x + 2{ }⊂ !2 Also, sketch this set. MATH 2362 Sample Exam 1 Page 7 of 7 7. Find a basis and the dimension for each space. a) W = a 0 0 a + c ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∈M22 : a,c ∈! ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ b) W = span 1+ 3x,2x + x2 ,1+ x − x2 ,−1+ x2 ,x + 2x2{ } Microsoft Word - exam2.sample.doc MATH 2362 SAMPLE EXAM 2 NAME: ________________________ STUDENT NUMBER: __________________ 1. There are 8 questions on 8 pages – CHECK YOURS NOW. 2. The time allotted to write the examination is 50 minutes. 3. Answer each question in the space provided – use the back of pages as needed. 4. Give exact answers where possible. 5. For full marks all relevant work should be included. 6. Good luck! Note, this exam is longer than the actual one and is intended to provide you with extra practice. 1 10 pts 2 8 pts 3 8 pts 4 5 pts 5 5 pts 6 6 pts 7 10 pts 8 10 pts Total 70 pts MATH 2362 Sample Exam 2 Page 1 of 8 1. Given is the matrix A = 1 3 −1 −1 2 2 6 −2 −2 4 1 0 0 1 2 1 0 0 2 −1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ a) Write down rref A( ) b) Find rank A( ) c) Find a basis for row A( ) d) Find a basis of col A( ) e) Find the nullity of AT f) Find a basis for the orthogonal complement of row A( ) g) If Ax = b , is b a vector from the column space, the row space, or neither? MATH 2362 Sample Exam 2 Page 2 of 8 2. Do one of the following two proofs. A. Show that if S = v1,…,v k{ } is a set of orthonormal vectors in ! k , then it must be a basis of ! k . B. Let B be a m× n matrix with orthonormal columns v1,…,v k . Show that m ≥ n and B TB = In . MATH 2362 Sample Exam 2 Page 3 of 8 3. For each statement decide whether True or False. Explain. [5] a) Every set of four vectors in ! 3 that span ! 3 is linearly independent. b) In ! n , if u ⋅v = 0 , then either u = 0 or v = 0 . c) If W1 and W2 are subspaces of the same vector space V , then either W1 =W2 or W1 ∩W2 = 0{ } . d) The orthogonal complement of the subspace with basis 1,2,3,4,0{ } in !5 must be 3-dimensional. e) If S is such that span S = ! d , then S has at least d vectors. (No explanation needed.) MATH 2362 Sample Exam 2 Page 4 of 8 4. Let B1 = 1,0,−1 , 0,1,1 , 0,0,1{ } be a basis for !3 and TB2→B1 = 1 3 2 2 0 1 0 −1 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ , v⎡⎣ ⎤⎦B2 = −5 0 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ be the transition matrix between B1 and B2 and a coordinate vector. [6] a) Find the basis B2 . b) Find v⎡⎣ ⎤⎦B1 and v . MATH 2362 Sample Exam 2 Page 5 of 8 5. Answer the following questions. [5] a) If A is a 4× 3 matrix, what is the largest possible dimension of its row space? Explain. b) If A is a 8×5 matrix, what is the smallest possible dimension of its null space? Explain. c) Find a set of 3 vectors in ! 3 that are dependent, but any subset of 2 vectors is linearly independent. d) For what values of a is the set 3+ x,2+ a2 + 3x{ } linearly independent? MATH 2362 Sample Exam 2 Page 6 of 8 6. a) Let V be the set of real numbers. Define vector addition by u⊕ v = 2u+ v and scalar multiplication by s⊗u = su . Is V a vector space? [3] b) Consider a subspace of ! 4 with a basis S = u1 = 1,2,0,−1 ,u2 = 2,0,−1,1 ,u3 = 1,1,0,0{ } Use Gram-Schmidt method to find an orthogonal basis