Definition: A square system is triangular if leading variable of the ith equation is the ith variable.
Recall that in a echelon system, if a variable is not a leading variable to any equation then it is called a free variable. Let’s justify this with an example.
Consider linear system given by
Here x1, x2, x4 are leading variables and x3, x5 are free variables. Notice that I can add set x3 and x5 to any number and they will be a unique solution. For example, x3 = 3 and x5 = 2. This gives the triangular system
which has a unique solution using back substitution. Let’s solve this linear system in general. We can set x3 = s1 and x5 = s2. Here s1, s2 are parameters that determine our solution space.
Work this out with class. Remark that the solution space is two dimensional.
With this technique, all echelon systems can be solved with back substitution if we ignore the constant equations like 0 = 0 or 1 = 0. We know how to handle the constant equations.
tldr: We can turn all linear systems into echelon systems without changing the solution space. We can organize the data of a linear system using matrices.
We can write a linear system as an augmented matrix. (give example in class).
Definitiion: The leading term of a row of a matrix is the leftmost nonzero term.
Definitiion: A matrix is in echelon form if * every leading term is in the column to the left of the leading term of the row below it * any zero rows are at the bottom
We can perform a series of elementary operations to turn a general linear system into a echelon system without changing the solution space.
The most important part about these operations is that they do not change the solution space. They do not change solution space. No changes to solution space. Solution space is the same. Same solution space.
Definition: The pivot positions are positions that contain a leading term. The pivot columns are columns that contain a pivot position. A pivot is the value of a pivot position.
Algorithm: Gaussian elimination is performed as follows: * find the pivot position in the first row * use elementary row operators to eliminate all value under the pivot position * continue
work out example in class
Definition: A matrix is in reduced echelon form if * it is in echelon form * all pivot positions contain a 1 * the only nonzero term in a pivot colum is in the pivot position
Algorithm: Gauss-Jordan elimination is performed as follows: * do Gaussian elimination * divide each row by the value of its pivot * eliminate all other values in pivot column.
work out example in class.
A linear system is homogenous if the numbers to the right of the equal sign are all zero. They always have the trivial solution