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Determine whether the statement below is true or false. Justify the answer.   The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.

Determine whether the statement below is true or false. Justify the answer.   The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 45E
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Determine whether the statement below is true or false. Justify the answer.
 
The homogenous equation Ax=0 has the trivial solution if and only if the equation has at least one free variable.
Determine whether the statement below is true or false. Justify the answer. The solution set of Ax = b is the set of all vectors of the form w = p + Vn, where v, is any solution of the equation Ax = 0. Choose the correct answer below. O A. The statement is false. The solution set could be empty. The statement is only true when the equation Ax = b is consistent for some given b, and there exists a vector p such that p is a solution. B. The statement is false. The solution set could be the trivial solution. The statement is only true when the equation Ax =b is inconsistent for some given b, and there exists a vector p such that p is a solution. C. The statement is false. The solution set could be empty. The statement is only true when the equation Ax = b is inconsistent for some given b, and there exists a vector p such that p is a solution. O D. The statement is true. The equation Ax = b is always consistent and there always exists a vector p that is a solution. O O O C
Transcribed Image Text:Determine whether the statement below is true or false. Justify the answer. The solution set of Ax = b is the set of all vectors of the form w = p + Vn, where v, is any solution of the equation Ax = 0. Choose the correct answer below. O A. The statement is false. The solution set could be empty. The statement is only true when the equation Ax = b is consistent for some given b, and there exists a vector p such that p is a solution. B. The statement is false. The solution set could be the trivial solution. The statement is only true when the equation Ax =b is inconsistent for some given b, and there exists a vector p such that p is a solution. C. The statement is false. The solution set could be empty. The statement is only true when the equation Ax = b is inconsistent for some given b, and there exists a vector p such that p is a solution. O D. The statement is true. The equation Ax = b is always consistent and there always exists a vector p that is a solution. O O O C
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