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:

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

Row reduced echelon form:

  1. Is in Echelon form
  2. All leading entries are 1
  3. 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 #

  1. Replace a row with the sum of itself and a multiple of another row
  2. Interchange/swap two rows
  3. Multiply a row with a nonzero constant

Solving linear systems #

Gaussian elimination for solving an augmented matrix:

  1. Leftmost nonzero column is the pivot column
  2. Select a nonzero entry is the pivot column as a pivot. If necessary swap rows to move the entry into the pivot position.
  3. Create zeros in all positions below the pivot by using row replacement operations.
  4. Cover the row with pivot position and above it. Repeat steps 1-3 until no rows are left. (Echelon form now)
  5. 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