{"id":40276,"date":"2025-06-27T15:28:30","date_gmt":"2025-06-27T15:28:30","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=40276"},"modified":"2025-06-27T15:28:31","modified_gmt":"2025-06-27T15:28:31","slug":"mark-each-statement-true-or-false","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/mark-each-statement-true-or-false\/","title":{"rendered":"Mark each statement True or False"},"content":{"rendered":"\n

Mark each statement True or False. Justify each answer. (a) A vector is any element of a vector space. (b) If
is a vector in a vector space V, then
is the same as the negative of
. (c) A vector space is also a subspace. (d)
is a subspace of
. (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)
,
, and
are in H, and (iii) c is a scalar and
is in H.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Here is the correct evaluation of each statement with justification:<\/p>\n\n\n\n

(a) A vector is any element of a vector space.<\/strong>
True<\/strong>
A vector space is defined as a set of vectors along with operations of addition and scalar multiplication that satisfy certain properties like associativity, distributivity, and the existence of a zero vector. By definition, any element belonging to the set of a vector space is called a vector. It is important to note that in mathematics, the term “vector” does not only refer to geometric arrows but applies to any element within a vector space, whether they are real numbers, functions, or matrices.<\/p>\n\n\n\n

\n\n\n\n

(b) If \ud835\udc63 is a vector in a vector space V, then \ud835\udc63 is the same as the negative of \ud835\udc63.<\/strong>
False<\/strong>
For any vector \ud835\udc63 in a vector space V, the negative of \ud835\udc63 is denoted by -\ud835\udc63. This is defined as the unique vector that satisfies the equation \ud835\udc63 + (-\ud835\udc63) = 0, where 0 is the zero vector of the space. Therefore, unless \ud835\udc63 is the zero vector itself, \ud835\udc63 is not the same as -\ud835\udc63. For example, if \ud835\udc63 is a nonzero vector, \ud835\udc63 and -\ud835\udc63 point in opposite directions in geometric terms.<\/p>\n\n\n\n

\n\n\n\n

(c) A vector space is also a subspace.<\/strong>
True<\/strong>
A subspace is defined as a subset of a vector space that is itself a vector space under the same operations. The entire vector space V is considered a subspace of itself because it trivially satisfies all the conditions required of a subspace, such as closure under addition and scalar multiplication, containing the zero vector, and so on.<\/p>\n\n\n\n

\n\n\n\n

(d) {0} is a subspace of V.<\/strong>
True<\/strong>
The set containing only the zero vector, {0}, is always a subspace of any vector space V. It contains the zero vector by definition, is closed under addition because 0 + 0 = 0, and is closed under scalar multiplication because c * 0 = 0 for any scalar c. This set is known as the trivial subspace.<\/p>\n\n\n\n

\n\n\n\n

(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) \ud835\udc62, \ud835\udc63, and \ud835\udc62 + \ud835\udc63 are in H, and (iii) c is a scalar and c\ud835\udc62 is in H.<\/strong>
False<\/strong>
The conditions provided are incomplete or miswritten. The correct conditions for a subset H to be a subspace of V are:<\/p>\n\n\n\n

    \n
  1. The zero vector of V is in H.<\/li>\n\n\n\n
  2. H is closed under addition, meaning for all \ud835\udc62, \ud835\udc63 in H, the sum \ud835\udc62 + \ud835\udc63 is also in H.<\/li>\n\n\n\n
  3. H is closed under scalar multiplication, meaning for all \ud835\udc62 in H and any scalar c, the product c\ud835\udc62 is in H.<\/li>\n<\/ol>\n\n\n\n

    The given wording incorrectly mentions \ud835\udc62, \ud835\udc63, and \ud835\udc62 + \ud835\udc63 being in H rather than stating closure under addition for any two elements. Therefore, the statement as written is false.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    Mark each statement True or False. Justify each answer. (a) A vector is any element of a vector space. (b) Ifis a vector in a vector space V, thenis the same as the negative of. (c) A vector space is also a subspace. (d)is a subspace of. (e) A subset H of a vector space […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-40276","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=40276"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40276\/revisions"}],"predecessor-version":[{"id":40278,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/40276\/revisions\/40278"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=40276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=40276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=40276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}