{"id":193548,"date":"2025-02-20T05:47:57","date_gmt":"2025-02-20T05:47:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=193548"},"modified":"2025-02-20T09:14:35","modified_gmt":"2025-02-20T09:14:35","slug":"the-set-of-all-first-degree-polynomial-functions-axb-a-not-equal-to-zero-whose-graphs-pass-through-the-origin-with-the-standard-operations-is-not-a-vector-space","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/20\/the-set-of-all-first-degree-polynomial-functions-axb-a-not-equal-to-zero-whose-graphs-pass-through-the-origin-with-the-standard-operations-is-not-a-vector-space\/","title":{"rendered":"The set of all first degree polynomial functions ax+b, a not equal to zero, whose graphs pass through the origin with the standard operations is not a vector space"},"content":{"rendered":"\n

The set of all first degree polynomial functions ax+b, a not equal to zero, whose graphs pass through the origin with the standard operations is not a vector space. State all of the axioms that fail and give a counterexample for each.<\/p>\n\n\n\n

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

The set of all first-degree polynomial functions of the form ( ax + b ), where ( a \\neq 0 ) and the graph passes through the origin, does not form a vector space. Let’s go through the axioms of a vector space and identify which ones fail.<\/p>\n\n\n\n

1. Axiom 1: Closure under addition<\/strong><\/h3>\n\n\n\n

A vector space must be closed under addition. This means if ( f(x) = ax + b ) and ( g(x) = cx + d ) are in the set, their sum ( f(x) + g(x) = (a + c)x + (b + d) ) must also be in the set.<\/p>\n\n\n\n

Failure:<\/strong>
The sum of two functions that pass through the origin may not pass through the origin. For example, ( f(x) = 2x ) and ( g(x) = 3x ) both pass through the origin, but their sum ( (2x + 3x) = 5x ) does not pass through the origin (as ( b \\neq 0 )).<\/p>\n\n\n\n

Counterexample:<\/strong>
Let ( f(x) = 2x ) and ( g(x) = 3x ). Their sum is ( f(x) + g(x) = 5x ), which does not pass through the origin.<\/p>\n\n\n\n

2. Axiom 2: Closure under scalar multiplication<\/strong><\/h3>\n\n\n\n

A vector space must be closed under scalar multiplication. If ( f(x) = ax + b ) is in the set, and ( k ) is a scalar, then ( k \\cdot f(x) = k \\cdot (ax + b) = k \\cdot ax + k \\cdot b ) must be in the set.<\/p>\n\n\n\n

Failure:<\/strong>
The scalar multiple of a function that passes through the origin may not pass through the origin. For example, ( f(x) = 2x ), and multiplying by a scalar ( k = 3 ) gives ( 3f(x) = 6x ), which still passes through the origin, but multiplying by a negative scalar (say, ( k = -1 )) could result in a function that doesn’t pass through the origin, depending on the value of ( b ).<\/p>\n\n\n\n

Counterexample:<\/strong>
Let ( f(x) = 2x ) and ( k = -1 ). Then ( k \\cdot f(x) = -2x ), which still passes through the origin, but if we add a constant term like ( b ), the result may not.<\/p>\n\n\n\n

Conclusion:<\/h3>\n\n\n\n

The primary failure is closure under addition, as the sum of two polynomials that pass through the origin may result in a polynomial that does not pass through the origin. Therefore, the set does not form a vector space.<\/p>\n\n\n\n

Now I will generate an image illustrating these examples and axioms.<\/p>\n\n\n\n

Here is a graph illustrating the examples for the vector space axioms that fail:<\/p>\n\n\n\n

    \n
  1. The first line passes through the origin.<\/li>\n\n\n\n
  2. The second line represents the sum of two lines that both pass through the origin, but the sum does not pass through the origin (failure of closure under addition).<\/li>\n\n\n\n
  3. The third line shows a scalar multiple of a line that passes through the origin, but may not always pass through the origin with other operations (failure of closure under scalar multiplication).<\/li>\n<\/ol>\n\n\n\n

    This image should help visualize the failures of the axioms. Let me know if you need further explanation!<\/p>\n\n\n\n

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

    The set of all first degree polynomial functions ax+b, a not equal to zero, whose graphs pass through the origin with the standard operations is not a vector space. State all of the axioms that fail and give a counterexample for each. The Correct Answer and Explanation is : The set of all first-degree polynomial […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-193548","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=193548"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193548\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=193548"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=193548"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=193548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}