Chapter 1 - Systems of Linear Equations

Definition. An equation in the unknowns x, y, z, . . . is called linear if both sides of the equation are a sum of (constant) multiples of x, y, z, . . ., plus an optional constant.

Definition. Let n be a positive whole number. We define $R^n$ = all ordered n-tuples of real numbers ( $x_1, x_2, ... ,$ $x^n$). An n-tuple of real numbers is called a point of $R^n$.$f(x) = x * e^{2 pi i \xi x}$

Each x represent an another dimension

1.2 Row Reduction

Use elimination method to reduce to the simplest form

Definition. A matrix is in row echelon form if:

1. All zero rows are at the bottom.

2. The first nonzero entry of a row is to the right of the first nonzero entry of the row above.

3. Below the first nonzero entry of a row, all entries are zero. Here is a picture of a matrix in row echelon form:

Definition. A pivot is the first nonzero entry of a row of a matrix in row echelon form.

Definition. A matrix is in reduced row echelon form if it is in row echelon form, and in addition:

1. Each pivot is 1

2. 1 pivot/number per column

1.3 Parametric Form

Definition. We say that x is a free variable if its corresponding column in A is not a pivot column.

Recipe: Parametric form. The parametric form of the solution set of a consistent system of linear equations is obtained as follows.

1. Write the system as an augmented matrix.

2. Row reduce to reduced row echelon form.

3. Write the corresponding (solved) system of linear equations.

4. Move all free variables to the right hand side of the equations.

If a system has one or more free variables, then it has infinite solution