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System of Linear Equations: Homogeneous Equation Video Lecture | Mathematics for Competitive Exams
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FAQs on System of Linear Equations: Homogeneous Equation Video Lecture - Mathematics for Competitive Exams
| 1. What is a homogeneous equation in a system of linear equations? |  |
Ans. In a system of linear equations, a homogeneous equation is one where all the constant terms on the right side of the equations are zero. It can be represented as Ax = 0, where A is the coefficient matrix and x is the variable vector.
| 2. How do you determine the number of solutions for a homogeneous system of linear equations? | |
Ans. The number of solutions for a homogeneous system of linear equations depends on the number of variables and the number of equations. If the number of variables is greater than the number of equations, then the system will have infinitely many solutions. If the number of variables is less than the number of equations, then the system will have a unique solution (trivial solution) where all variables are zero.
| 3. Can a homogeneous system of linear equations have a non-trivial solution? | |
Ans. Yes, a homogeneous system of linear equations can have a non-trivial solution. A non-trivial solution is a solution where at least one variable is not equal to zero. If the system has infinitely many solutions, then there will be infinitely many non-trivial solutions.
| 4. What is the relationship between the homogeneous equation and the trivial solution? | |
Ans. The trivial solution in a homogeneous equation is when all variables are equal to zero. In other words, it is the solution where no variable has a non-zero value. The trivial solution always exists in a homogeneous equation because substituting zero for all variables satisfies the equation.
| 5. How can a homogeneous system of linear equations be solved? | |
Ans. To solve a homogeneous system of linear equations, one can first write the system in matrix form as Ax = 0, where A is the coefficient matrix and x is the variable vector. Then, using techniques such as row operations or matrix inverse, the reduced row echelon form of the augmented matrix can be obtained. The solutions can be determined by expressing the variables in terms of free variables in the system.
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Nov 22, 2024 Last updated |
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