{"id":47240,"date":"2025-07-02T11:15:43","date_gmt":"2025-07-02T11:15:43","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=47240"},"modified":"2025-07-02T11:15:45","modified_gmt":"2025-07-02T11:15:45","slug":"in-exercises-23-and-24-mark-each-statement-true-or-false","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/in-exercises-23-and-24-mark-each-statement-true-or-false\/","title":{"rendered":"In Exercises 23 and 24, mark each statement True or False."},"content":{"rendered":"\n

In Exercises 23 and 24, mark each statement True or False. Justify each answer. 1. A homogeneous equation is always consistent. 2. The equation Ax = 0 gives an explicit description of its solution set. 3. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. 4. The equation x = p + tv describes a line through a point parallel to p. 5. The solution set of Ax = b is the set of all vectors of the form p + ya, where ya is any solution of the equation Ax = 0. 6. A homogeneous system of equations can be inconsistent. 7. If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. 8. The effect of adding p to a vector is to move the vector in the direction parallel to p. 9. The equation Ax = b is homogeneous if the zero vector is a solution.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

Here are the correct answers and justifications for each statement.<\/p>\n\n\n\n

Exercise 23<\/strong><\/p>\n\n\n\n

    \n
  1. True.<\/strong>\u00a0A homogeneous equation is of the form Ax = 0. It is always consistent because it has at least one solution, the trivial solution, where x is the zero vector. Since A times the zero vector always results in the zero vector, this solution always exists.<\/li>\n\n\n\n
  2. False.<\/strong>\u00a0The equation Ax = 0 is an implicit description of the solution set. It defines the condition that any solution vector x must satisfy. An explicit description is what you find after solving the system, typically by writing the solution in parametric vector form (e.g., x = t*v for some vector v), which explicitly generates all possible solutions.<\/li>\n\n\n\n
  3. False.<\/strong>\u00a0A homogeneous equation Ax = 0\u00a0always<\/em>\u00a0has the trivial solution, regardless of whether there are free variables. The presence of a free variable indicates that the system has\u00a0nontrivial<\/em>\u00a0solutions (solutions other than the zero vector) in addition to the trivial one. The statement incorrectly connects the existence of the trivial solution to the presence of free variables.<\/li>\n\n\n\n
  4. False.<\/strong>\u00a0The equation x = p + t*v describes a line that passes through the point corresponding to the vector\u00a0p<\/strong>\u00a0and is parallel to the direction vector\u00a0v<\/strong>. The statement incorrectly claims the line is parallel to\u00a0p<\/strong>.<\/li>\n\n\n\n
  5. True.<\/strong>\u00a0This is a fundamental theorem in linear algebra. If the equation Ax = b is consistent and has a particular solution\u00a0p<\/strong>, then the full solution set consists of all vectors of the form\u00a0p<\/strong>\u00a0+\u00a0v_h<\/strong>, where\u00a0v_h<\/strong>\u00a0is any solution to the corresponding homogeneous equation Ax = 0. This means the solution set of Ax = b is a translation of the solution set of Ax = 0.<\/li>\n<\/ol>\n\n\n\n

    Exercise 24<\/strong><\/p>\n\n\n\n

      \n
    1. False.<\/strong>\u00a0This statement is the opposite of statement 1. A homogeneous system Ax = 0 is always consistent because the trivial solution (x = 0) is always a possible solution. An inconsistent system is one with no solutions at all.<\/li>\n\n\n\n
    2. False.<\/strong>\u00a0A nontrivial solution is any nonzero vector x that solves the equation. For x to be nonzero, it only needs to have at least one nonzero entry; the other entries can be zero. For example, x = [1, 0] is a nontrivial vector and could be a solution to some homogeneous system.<\/li>\n\n\n\n
    3. True.<\/strong>\u00a0Vector addition can be interpreted geometrically. Adding a vector\u00a0p<\/strong>\u00a0to another vector\u00a0v<\/strong>\u00a0results in a new vector\u00a0v<\/strong>\u00a0+\u00a0p<\/strong>. This operation translates the endpoint of the vector\u00a0v<\/strong>\u00a0by the displacement vector\u00a0p<\/strong>. The movement is in the direction and magnitude of\u00a0p<\/strong>.<\/li>\n\n\n\n
    4. True.<\/strong>\u00a0An equation is defined as homogeneous if it is in the form Ax = 0. If the zero vector is a solution to the equation Ax = b, we can substitute x = 0 into the equation. This gives A(0) = b, which simplifies to 0 = b. Therefore, the vector b must be the zero vector, and the equation is Ax = 0, which is homogeneous by definition.<\/li>\n<\/ol>\n\n\n\n
      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      In Exercises 23 and 24, mark each statement True or False. Justify each answer. 1. A homogeneous equation is always consistent. 2. The equation Ax = 0 gives an explicit description of its solution set. 3. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least […]<\/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-47240","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47240","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=47240"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47240\/revisions"}],"predecessor-version":[{"id":47262,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/47240\/revisions\/47262"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=47240"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=47240"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=47240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}