# Linear Systems

## Echelon forms #

**Leading entry**: first nonzero element of a row.

A matrix is in **Echelon form** if it satisfies the following three properties:

- All nonzero rows are above all zero rows.
- Each leading entry of a row is in a column to the left of the leading entry of the row below.
- All entries in a column below a leading entry are zero.

**Row reduced echelon form**:

- Is in Echelon form
- All leading entries are 1
- Each leading entry is the only nonzero entry in its
*column*.

## Gaussian elimination #

**Augmented matrix**: can be used to solve a system of equations. Every row represents a variable and the last row the value.

### Elementary row operations #

- Replace a row with the sum of itself and a multiple of another row
- Interchange/swap two rows
- Multiply a row with a nonzero constant

### Solving linear systems #

Gaussian elimination for solving an augmented matrix:

- Leftmost nonzero column is the pivot column
- Select a nonzero entry is the pivot column as a pivot. If necessary swap rows to move the entry into the pivot position.
- Create zeros in all positions below the pivot by using row replacement operations.
- Cover the row with pivot position and above it. Repeat steps 1-3 until no rows are left. (Echelon form now)
- Create zeros above each pivot by going from bottom right to top left. (Reduced echelon form now)

## Solutions #

**Free variable**: if there is no corresponding pivot column

No free variables -> unique solution If at least one free variable -> infinitely many solutions

**Consistent** linear system: iff the rightmost column of the augmented matrix is not a pivot column.

Otherwise **inconsistent** and there are no solutions (creates a contradiction)

## Solution sets & linear independence #

**Homogeneous** linear system: the system can be written in the form $A\vec{x}=\vec{0}$ where $A$ is a matrix and $\vec{0}$ the zero vector.

- Has always a
**trivial solution**when all variables are zero ($\vec{x} = \vec{0}$) - Has a
**nontrivial solution**iff the equation has at least one free variable - The columns of matrix $A$ are
**linearly independent**iff a vector equation has only the trivial solution

**Parametric form**: solution written as a sum of vectors.

## Relations #

Linearly dependent $\Leftrightarrow$ one of the vectors is a multiple of another (a linear combination) $\Leftrightarrow$ has a nontrivial solution $\Leftrightarrow$ determinant is zero

Linearly independent $\Leftrightarrow$ has only the trivial / unique solution $\Leftrightarrow$ the determinant is non-zero