Linear

Definition 1.1.2: Matrix

an 𝑚 ×𝑛 array of real numbers with 𝑚 rows and 𝑛 columns:

Definition 1.1.3: Euclidean Vector

an ordered list of real numbers that can be represented as a point in n-dimensional space.

Definition 1.1.7: Linear Equation

an equation of the variables 𝑥𝑖 of the form

𝑎1⁢𝑥1 + 𝑎2⁢𝑥2 + ⋯ + 𝑎𝑛⁢𝑥𝑛 = 𝑏.
whose solution is a is a Euclidean Vector

Definition 1.1.9: System of Linear Equation (Linear System)

Definition 1.1.12: Consistent v. Inconsistent

A linear system is consistent if its solution set is non-empty (that is, there exists a solution for the system). Otherwise it is inconsistent.

Definition 1.1.22: Augmented Matrix and Coefficient Matrix

Definition 1.2.2: Equivalent

Two systems of linear equations (and their corresponding augmented matrices) are said to be equivalent if they have the same solution set.

Definition 1.2.5: Row Operations

operations performed on the rows of a matrix to simplify it without changing the solution set of the corresponding system of linear equations.

Definition 1.2.8: Reduced Row Echelon Form (RREF)

a simplified form of a matrix obtained through elementary row operations that satisfies four conditions:

1. Leading 1s: The first nonzero entry in each nonzero row is 1 (called a pivot).

2. Pivot positions: Each pivot is to the right of the pivot in the row above it.

3. Zero rows: Any rows consisting entirely of zeros are at the bottom of the matrix.

4. Pivot columns: Each pivot is the only nonzero entry in its column (all entries above and below it are zero).

Definition 1.4.3: Bound and Free Variables

Bound variables are variables corresponding to pivot columns in a matrix's RREF. Their values are determined by the equations and may depend on the free variables.

Free variables are variables corresponding to columns without pivots (the rest) in a matrix's RREF. They can take any real value (assuming the system is consistent) and are used to express the bound variables.

Definition 2.1.3: Linear Combination

A vector formed by multiplying a set of vectors by scalars (real numbers) and adding the resulting vectors together.

Definition 2.1.4: Span

the span of a set of vectors is the collection of all linear combinations of that set