Chapter 1 Section 1.3 Consistent Systems of Linear Equations.

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Presentation transcript:

Chapter 1 Section 1.3 Consistent Systems of Linear Equations

⅓R 1 R 1 +R 2 -2R 1 +R 3 ½R 2 2R 2 +R 3 We can rewrite this so that the variables whose coefficient is the leading 1 in row of the matrix in reduced echelon form is on the right side of the equation and all other variables are on the left. General Solution Particular Solution

Dependent vs Independent Variables The reduced echelon form of an augmented matrix for a consistent system of equations will never have a leading 1 in the last (or augmented) column. Each of the non-augmented columns corresponds to a variable in the system. Each of the columns with a leading 1 the corresponding variable appears on right side when the general solution is expressed. The variables that correspond to the columns with the leading 1’s are the dependent (constrained or determined) variables. The variables that correspond to the columns without the leading 1’s are the independent (unconstrained or free) variables.

Examples Each of the matrices below is the augmented matrix for a system of equations. Given the general solution for the system in terms of the dependent variables and tell which variables are dependent and which are independent. Matrix VariablesGeneral Solution No Solution the system is not consistent

Solutions to linear systems The number of solutions to a linear system only has three possible numbers; None, a unique (only one) solution, or an infinite number of solutions. 1.The system has none when it is inconsistent. 2.The system has a unique solution when it is consistent but has no independent variables (i.e. all variables are dependent). 3.The system has infinitely many solutions when it consistent but has at least one (maybe more) independent variables. Inconsistent No Solutions No Independent Variable Homogeneous Systems of Equations A homogeneous system of linear equations is a system where all the constants are zero. In matrix form the augmented (or last) column is all zeros. Because of this a homogeneous system is always consistent. It always has the solution of having all variables being zero. If there are no independent variable this solution is unique (only one) if there is an independent variable there is an infinite number of solutions.