Problem #1: Let V be the set of all ordered pairs of real numbers (111, 142) with 142 > 0. Consider the following addition and scalar multiplication operations on u : (ul, H2) and v : (v1, v2): 11 + V : (111 + v1 * 5, 3u2v2), ku : (kill, kuz) Use the above operations for the following parts. (3) Compute u + v for u : (6, 7) and V : (*6, 7). (b) If the set V satisfies Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector? (c) Ifu : (*6, 4), what would be the negative ofthe vector u referred to in Axiom 5 ofa vector space? (Don't forget to use your answer to part (b) here!) Problem #1(a): |:| enter your answer in the form a,b Enter your answer symbolically, _ Problem #1(b): as in these examples enter your answer In the form a,b Enter our answers mbolicall , . Problem #1(c): :I as in these example/s y enter your answer In the form a,b Just Save Submit Problem #1 for Grading Problem # 2: Which of the following sets are closed under scalar multiplication? (i) The set of all vectors in R2 of the form (a, b) Where a + 2b = 0. (ii) The set of all 3 X 3 matrices Whose trace is equal to 4. (The trace of a square-matrix is the sum of the diagonal entries.) (iii) The set of all polynomials in P2 of the form a0 + all x + a2 x2 Where the product a0 a1 a2 2 O. (A) none of them (B) all of them (C) (i) and (iii) only (D) (i) and (ii) only (E) (ii) only (F) (iii) only (G) (ii) and (iii) only (H) (i) only Problem #2: l Just Save l I Submit Problem #2 for Grading Problem #2 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #3: Let O A = U U 8 0 (a) Find a basis for the column space of A. (b) Find a basis for the nullspace of A. (A) {(1, 0, 0, 0), (0, 1, 0, 0); (B) { (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1); (C) {(1, 5, 1)} (D) { (1, 5, 1), (0, 5, 0); (E) {(1, 5, 1), (0, 5, 0), (1, 7, 0), (1, 1, 1); (F) {(0, 5, 0); (G) {(1, 1, 1); (H) { (1, 5, 1), (0, 5, 0), (1, 7, 0)} Problem #3(a): Select v (A) {(-1, , 1, 0), (-1, 5, 0, 1)} (B) {(-1, 2, 0, 1); (C) {(-1, -2, 1, 0), (-1, 2, 0, 1); (D) {(-1, ", 0, 1) ; (E) {(-1, -5, 1, 0), (-1, 2, 0, 1); (F) { (-1, 2, 0, 1); (G) ((-1, 2, 1, 0), (-1, 2, 0, 1); (H) (-1, 5, Problem #3(b): Select v Just Save Submit Problem #3 for Grading Problem #3 Attempt #1 Attempt #2 Attempt #3 Your Answer: 3(a) 3(a) 3(a 3 ( b ) 3 (b ) 3 ( b ) Your Mark: |3(a) 3(a) 3(a) 3 (b ) 3 (b ) 3 ( b )Problem #4: Consider the following statements. (i) If V is a vector space then there exists an x E I such that x + v = v for all v E V. (ii) M2 x 2 is a subspace of M3 x 3 (iii) A vector space is a subspace of itself. Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True, False, False, then you would enter '1,2,2' into the answer box below (without the quotes). Problem #4: Just Save Submit Problem #4 for Grading Problem #4 Attempt # 1 Attempt #2 Attempt # 3 Your Answer: Your Mark: Problem #5: Consider the matrices N 0 21 -2 -16 O A = b = -56 w -28 -21 W O 8 -28 w Which of the following statements is true? (A) b is in the column space of A, but not the nullspace of A (B) b is in both the column space of A and the nullspace of A (C) b is not in the column space of A and b is not in the nullspace of A (D) b in the null space of A, but not the column space of A Problem #5: Select v