There’s an upshot to the whole starting-at-weird-times my MWF classes have. As long as I keep forgetting that the class starts at 9:10 rather than 9:00, I’m always very early. On the other hand, this works against me for the 1:50 class.
On to the math.Section 2.3
Objectives:
- Analyze the reduced row echelon form of an augmented matrix
- Solve a system of m linear equation containing n variables
- Express the solution of a system with an infinite number of solutions
In any set of linear equations, there exists either no possible solution, exactly one possible solution, or infinitely many solutions. It’s irrelevent how many equations or variables are involved.
If the system is inconsistent (no solutions), reduced row echelon form will show a set of coefficients which cannot generate the constant for that equation. Usually this means 0 will be shown with equality to a nonzero number. For instance, [0 0 0|1]. If a row like that appears in a matrix, the matrix is inconsistent. An equation capable of producing such a row would have no solution.
If a system of equations has more equations than variable, either one of the equations is redundant, or the system is inconsistent. If the system is consistent, normally one will find that the process of rendering the matrix to reduced row echelon form will convert the final row of equations to a full string of zeros. This sort of system is referred to as Overdetermined. Overdetermined systems tend to be inconsistent more often than non-overdetermined systems.
If the system of equations has fewer equations than unknowns, the system is known as Underdetermined. These sorts of systems frequently give infinitely many solutions.