Linear Algebra Definitions

Elementary row operations

Basic operations that can be performed on the rows of a matrix to simplify it.

Row Echelon Form

A form of a matrix where all non-zero rows are above any rows of all zeros and leading coefficients are to the right of the leading coefficients of the previous row.

Reduced Row Echelon Form

A form of a matrix where it is in row echelon form and all leading coefficients are 1 and are the only non-zero entries in their columns.

Coefficient matrix

The matrix that consists of the coefficients of the variables in a system of linear equations.

Augmented matrix

A matrix that contains the coefficients of a linear system and the constants from the equations.

Consistent system

A system of equations that has at least one solution.

Inconsistent system

A system of equations that has no solutions.

Basic variable

Variables that correspond to the leading 1's in the row echelon form of a matrix.

Free variable

Variables that are not leading variables in a system of equations, which can take on any value.

Homogeneous system

A system of equations that has all zero constant terms.

Non-homogeneous system

A system of equations that has at least one non-zero constant term.

Parametric Vector Form

A representation of a vector in terms of parameters that allow for different solutions.

Linear combination

An expression formed by summing scalar multiples of vectors.

Trivial solution

The solution to a homogeneous system where all variables are equal to zero.

Non-trivial solution

A solution to a homogeneous system where at least one variable is not equal to zero.

Linearly independent vectors

Vectors that cannot be expressed as linear combinations of each other.

Linearly dependent vectors

Vectors that can be expressed as linear combinations of each other.

Span of a set of vectors

The set of all possible linear combinations of a given set of vectors.

Transformation

A function that maps elements from one set into another.

Domain

The set of all possible input values for a function.

Co-domain

The set that contains all potential output values of a function.

Range

The set of actual output values of a function.

Linear transformation

A transformation that preserves vector addition and scalar multiplication.

Onto

A function where every element in the co-domain is the image of at least one element from the domain.

One-to-One

A function where each element in the domain maps to a unique element in the co-domain.