MATH 218 Midterm, Summer 2014

1 June 23, 2014 MATHEMATICS 218 NAME. ------------------- Time : 65 minutes Midterm Exam ID# ------------------ Summer 2013-14 PROBLEM GRADE PART I 1 ------- /14 2 ------- /16 3 ------- / 20 4. ------- / 10 PART II 5 6 7 8 9 a a a a a b b b b b c c c c c d d d d d e e e e e 5-9 ------- / 20 PART III Answer True or False only in the table below: A B C D E F G H I J 10 ------- / 20 TOTAL ------- / 100 2 PART I. Answer each of the following problems in the space provided for each problem (Problem 1 to Problem 4). 1. Find the values of a and b for which the system x -2y -z = 5 3x -7y - az = 0 -2x +3y + z = b has a. no solution b. a unique solution c. infinitely many solutions. [ 14 points] 3 2. Let A= ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 4 1 2 1 3 1 1 1 1 0 1 0 2 1 0 1 (a) Find a basis of the null space N(A). (b) Find a basis of the column space Col(A). [ 16 points] 4 3. a. Let V be the subset of R 3 defined by: V= ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ∈ ∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − R x y R y x y x , 2 3 Show that V is a subspace of R 3. Find a basis for V. [ 10 points] 5 3b. Let W be the subspace of R 3 given by W= ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ∈ ∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − R a b R b a a , 2 3 (Do not show W is a subspace) Find a basis for V∩W . [ 10 points] 4. Let V be a vector space with basis { v1, v2,…, vn } . Show that every vector v∈V can be written uniquely as a linear combination v= c1v1 + c2v2 + … + cnvn [ 10 points] 6 PART II. Circle the correct answer for each of the following problems ( Problem 5 to Problem 9) IN THE TABLE OF THE FRONT PAGE . [4 points for each correct answer]. 5. The value of the number k for which the matrix A= ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 2 0 1 1 3 2 1 k k , is not invertible is: (a) k= 2 (b) k= 0 (c) k= -1/4 (d) k= 4 (e) none of the above [ 4 points] 6. Let V= 0} (0) ( ) ( ) { 3 = ′′ | ∈ p P R p x Then dim V= (a) 1 (b) 2 (c) 3 (d) 4 (e) none of the above. [ 4 points] 7. Which one of the following statements is FALSE: (a) Any set of 6 polynomials in P4 is linearly dependent. (b) Any linearly independent set of 5 polynomials in P4 is a basis of P4. (c) Any set of 5 polynomials in P4 that spans P4 is a basis of P4. (d) Any set of 6 polynomials in P4 spans P4. (e) none of the above. [ 4 points] 7 8. The subspace of R3 spanned by S= ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − , 3 1 2 3 1 1 , 0 2 1 has dimension: (a) 1 (b) 2 (c) 3 (d) 4 (e) none of the above. [ 4 points] 9. Let {u, v, w} be a basis for a vector space V. Let U=Span {u}, W=Span {v,w}, which one of the following statements is FALSE (a) {2u, v, 3w} is a basis for V. (b) V=U+W (c) {u+w, u+v+w, v}is basis of V (d) U∩W={0} (e) None of the above [ 4 points] PART III. Answer TRUE or FALSE only ( 3 points for each correct answer, 0 for no answer, and -1 point for each wrong answer): [ 20 points] A. The set of all 2×2 invertible matrices is a subspace of M 2 2× . B. The set of polynomials {2+x, 3+4x3, x+x3, 1+x+x2 , x2 } is linearly independent in P3. C. Let W= ⎪⎭ ⎪⎬ ⎫ ⎪⎩ ⎪⎨ ⎧ ∈ ∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ R a b R b a , 1 3 . Then W is a subspace of R3. D. Let W= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∈ ⎟⎟∈ ⎠ ⎞ ⎜⎜ ⎝ ⎛ × R a b M b b a , 3 0 2 2 . Then W is a subspace of M 2 2× of dimension 2. E. The space of all 3×3 upper triangular matrices has dimension 6. 8 F. Let U be the subspace of P3 given by U= 0} ( ) 0 (0) ( ) ( ) { 3 2 3 = ′′ = | ∈ + + + = x and p p P R d cx bx ax p x . Then dim U=2. G. If A, B, and C are n×n invertible matrices over R such that AB=AC, then B=C. H. The set of all solutions of a nonhomogeneous system of m linear equations in n unknowns is a subspace of Rn. I. If A and B are n×n matrices such that T ABA is invertible, then A must be invertible. J. If B is a 3×3 matrix such that B BT = − , then B is not invertible [20 points]