Table of Contents
What do we mean by trivial solution?
“Trivial” can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the differential equation.
What are trivial and nontrivial solutions?
Since the zero solution is the “obvious” solution, hence it is called a trivial solution. Any solution which has at least one component non-zero (thereby making it a non-obvious solution) is termed as a “non-trivial” solution.
What does trivial solution mean in matrices?
The solution x = 0 is called the trivial solution. A solution x is non-trivial is x = 0. • The homogeneous system Ax = 0 has a non-trivial solution if and only if the equation has at least one free variable (or equivalently, if and only if A has a column with no pivots).
Does trivial solution mean no solution?
The system of equation in which the determinant of the coefficient is zero is called non-trivial solution. And the system of equation in which the determinant of the coefficient matrix is not zero but the solution are x=y=z=0 is called trivial solution.
Is trivial solution linearly independent?
If you get only the trivial solution (all coefficients zero), the vectors are linearly independent. If you get any solution other than the trivial solution, the vectors are linearly dependent.
How do you tell if a matrix has a nontrivial solution?
If the system has a solution in which not all of the x1,⋯,xn are equal to zero, then we call this solution nontrivial . The trivial solution does not tell us much about the system, as it says that 0=0!
Does the equation have a nontrivial solution?
Answer: False. If x is not equal to the zero vector, and Ax = 0, then x is a nontrivial solution….Answers to Quizlet 1-5.
| Question 6. A is a 3×3 matrix with 3 pivot positions. Select all the statements which must be true for this A. | Ax = 0 has a nontrivial solution. | False |
|---|---|---|
| Ax = b has at least one solution for every possible b. | True |
How do you know if a matrix has a trivial solution?
A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.
What has only the trivial solution?
Thus, for homogeneous systems we have the following result: A nxn homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions.
Is trivial solution dependent?
What is a trivial linear combination?
Definition: A linear combination a1v1 + + anvn is called trivial if all the a’s are zero. Otherwise it is nontrivial. Definition: a set of vectors is called linearly independent if the only linear combination of them that adds to 0 is the trivial combination.