Question

Determine whether each statement is True or False. Justify each answer. a. A vector is any element of a vector space. Is this statement true or false? O A. True by the definition of a vector space O B. False; not all vectors are elements of a vector space. O C. False; a vector space is any element of a vector. b. If u is a vector in a vector space V, then (-1) is the same as the negative of u. Is this statement true or false? O A. True because for each u in V, - u =(-1) O B. False because for each u in V, there is a vector - u in V such that u +(-u) = 0 O C. True because for each u in V, there is a vector - u in V such that u +(-u) = 0 OD. False because for each u in V, - uru c. A vector space is also a subspace of itself. Is this statement true or false? O A. False because the axioms for a vector space do not include all the conditions for being a subspace OB. False because the conditions for a subspace do not include all the axioms for being a vector space O C. True because the axioms for a vector space include all the conditions for being a subspace O D. True because the conditions for a subspace include all the axioms for being a vector space d. R2 is a subspace of R. Is this statement true or false? O A. True because R2 contains the zero vector, and is closed under vector addition and scalar multiplication OB. False because R2 is not even a subset of R$ O C. False because R3 is not even a subset of R2 OD. True because R3 contains the zero vector, and is closed under vector addition and scalar multiplication e. A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u + v are in H, and (iii) c is a scalar and cu is in H. Is this statement true or false? O A. False; these conditions are stated correctly, however there is at least one additional condition. O B. False; parts (ii) and (iii) should state that u and v represent all possible elements of H. O C. False; part (i) is not required. O D. True; this is the definition of a subspace.

Answer 1

. (a.) A vector is any element of a vector space - TRUE by the definition of a vector space. . (6.) q u is a vector space in a vector space V, then (-1) is the same as the negative of u. ❤ TRUE because for each u in Vi -U = (-1) i obtion (A:) v . (6.) A vector space is also a subsace of itself. - TRUE because the asions for a vector space include all the conditions for being a subspace. (: option (c)~) (d) TR is a subspace of TR3 = FALSE, because that is NOT even a subset of TR3. (: option (8) ~) I Elements of Ra are (a,b), ie... 2-tuple but elements of its are (a, b, c), 3- teple) Cs Scanned with ..TR is not even a subset of 7p3 CamScanner

(e.) SE: Atarra F Ginen stame statement in parts (ii) and (iii) should u and v represent all of H. (i. Oblrón (8.) v) FALSE: state that possible elements cs Scanned with CamScanner