Exploring Existence and Uniqueness of Solutions in Linear Systems of Equations

Two fundamental questions about a linear system involve existence and uniqueness

In mathematics, when it comes to a linear system of equations, two fundamental questions arise: existence and uniqueness

In mathematics, when it comes to a linear system of equations, two fundamental questions arise: existence and uniqueness.

1. Existence:
The first question is whether a solution to the linear system exists or not. This means we are seeking to determine whether there is at least one combination of values for the variables that satisfies all the given equations simultaneously.

To check for existence, we can examine the coefficient matrix of the system. This matrix is formed by extracting the coefficients of the variables from each equation. If the coefficient matrix is invertible (i.e., its determinant is not zero), then a solution exists. This is known as the existence theorem.

2. Uniqueness:
The second question relates to the uniqueness of the solution. Once we establish that a solution exists, we want to determine if it is the only solution or if there are multiple solutions. A unique solution implies that there is only one combination of values for the variables that satisfies all the equations in the system.

To check for uniqueness, we examine the coefficient matrix along with the augmented matrix formed by adding the constant terms of the equations. If the coefficient matrix is invertible and the augmented matrix does not have any free variables (variables that can take any value), then the solution is unique. This condition is known as the uniqueness theorem.

It is important to note that these theorems apply to systems of linear equations where the number of equations equals the number of variables. If the system is underdetermined (fewer equations than variables), there will be infinitely many solutions. If the system is overdetermined (more equations than variables), there may be no solution at all.

In summary, the existence and uniqueness of solutions in a linear system are determined by checking the invertibility of the coefficient matrix and the presence of any free variables or extra equations.

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