hello I need help solving those questions please

'oblem #7: Which of the following statements are always true for vectors in R3? (i)Ifu-(vxw)=4thenw'(vxu)=-4 (ii) (7n + v) x (u + 4v) = 29 (u x v) (iii) If u is orthogonal to v and w then u is also orthogonal to Ilwll v + \"VII 11 (A) (iii) only (B) (ii) only (C) (i) and (ii) only (D) none of them (E) all of them (F) (i) only (G) (ii) and (iii) only (H) (i) and (iii) only Problem #7: Select 0 Just Save Your work has been saved! (Back to Admin Page) Submit Problem #7 for Grading Problem #7 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: 'oblem #8: Let V be the set of all ordered pairs of real numbers (u1,u2) with 142 > 0. Consider the following addition and scalar multiplication operations on u = (:41, 42) and v = (v1, v2): 11 + v =(u1+ v1+ 6,8u2v2), ku =(ku1,ku2) Use the above operations for the following parts. (3) Compute u + v for u = (7,6) and v = (3, 8). (b) If the set V satises Axiom 4 of a vector space (the existence of a zero vector), what would be the zero vector? (C) If u = (8, 5), what would be the negative of the vector u referred to in Axiom 5 of a vector space? (Don't forget to use your answer to part (b) here!) Problem #8(a): enter your answer in the form a,b Enter your answer symbolically, _ Problem #8(b): as in these examples enter your answer In the form a,b P bl #8 . Enter your answer symbolically, _ ro em (c). as in these examples enter your answer In the form a,b 'roblem # 9: 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 6b = 0. (ii) The set of all 3 x 3 matrices whose trace is equal to 1. (iii) The set of all polynomials in P2 of the form a0 + a1): + a2 x2 where the product a0 a1 a2 5 0. (A) (ii) and (iii) only (B) (iii) only (C) (i) only (D) (i) and (iii) only (E) (i) and (ii) only (F) none of them (G) all of them (H) (ii) only Problem #9: Select 8 Just Save Your work has been saved! (Back to Admin Page). Submit Problem #9 for Grading 'roblem # 1 0: Which of the following sets are closed under addition? (i) The set of all vectors in R2 of the form (a, b) where b = a2. (ii) The set of all 2 x 2 matrices that have the vector [3 2]T as an eigenvector. (iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where :10 = a2. (A) (ii) and (iii) only (B) (i) and (ii) only (C) (i) and (iii) only (D) (iii) only (E) none of them (F) (i) only (G) (ii) only (H) all of them Problem #10: Select )blem #1 1: Let v1 = (3, 2, 3) and v2 = (3, 1,2). Which of the following vectors are in span{v1,v2}? (i) (6,1,0) (ii) (9,0,2) (iii) (15,1,1) (A) (ii) only (B) (i) and (ii) only (C) all of them (D) none of them (E) (i) only (F) (i) and (iii) only (G) (iii) only (H) (ii) and (iii) only Problem #11: Select 9 i) For vector (6, 1, 0): 61(3, 2, 3) l (32(3, 1,2) : (6,1, 0) This gives us the following system of equations: 361 362 I 6 261 +02 : 1 3(31 262 : 0 Now, you can solve this system. You can use various methods such as substitution, elimination, or matrices. Here, I'll use substitution: 2 From the third equation, we get (:1 : :362. Substitute this into the first equation: 3 (3.22) 3C2 : 6 Solving this, you get (:2 : 1, and then cl : 1. So, (6, 1, 0) is in the span of '01 and '02. ii) For vector (9, 0, 2): 61(3, 2, 3) + (32(3, 1, 2) = (:9, D, 2) This gives the system: 301 3C2 : 9 261 +62 : 0 3(31 262 : 2 This system has no solution because the first and third equations are inconsistent (they cannot both be true at the same time). Therefore, (:9, O, 2) is not in the span of U1 and '02. iii) For vector (15, 1, 1): (31(3, 2, 3) + 2(3,1,2) : (15,1, 1) This gives the system: 301 302 : 15 261 + C2 : 1 3(31 262 = 1 Solving this system, you find (:1 : 5 and (:2 : 2, so (15, 1, 1) is in the span of U1 and '02. In summary, solving the system of equations is a way to determine whether a vector is in the span of a set of vectors. If the system has a solution, the vector is in the span; if not, it is not in the span