Linear Algebra Definitions

Rank

number of linearly independent equations

number of nonzero rows after Gaussian elimination

Echelon Form

1. The first nonzero entry in any row that occurs to the right of the 1st nonzero entry in the previous row

2. All zero rows are at the bottom of the matrix

Row Reduced Echelon Form (RREF)

1. Matrix in Echelon form

2. Every entry above a pivot is 0

3. All pivots are 1

Subspace of V

a nonempty subset w if a vector space if w is closed under linear combos

Column Space

The span of the columns of M given the matrix M E M(m,n)

Span

the set of all possible linear combinations that can be formed by scaling and adding those vectors together

AX=0

Homogeneous system of linear equations where x=o is always a solution

Linearly Independent Vectors

The only way to combine the vectors to get the zero vector is by setting every scalar multiplier to zero

Linearly Dependant Vectors

contains at least one redundant vector that can be written as a linear combination of the others

Nullspace(A)

The set of solutions to AX=0 in R^m

Pivot Columns

Columns that contain the pivots in RREF

Basis

A linearly independent set, which spans a vector space

Dimension of a vector space

number of elements in a basis

Rowspace

the set of all possible linear combinations of its row vectors

Nullity(A)

Dimension of Nullspace(A)

Nonsingular A

Ax=B has a unique solution for all B E R^m