Rank

Introduced in Math 115

[!Definition 2.3.1]

The rank of matrix A, denoted by rank(A), is the number of leading entries in any REF of A.

If [A | $\vec{b}$] is an augmented matrix, then rank([A | $\vec{b}$]) is the number of leading entries in any REF of [A | $\vec{b}$]

rank(A) $\le$ min{m, n}

[!System Rank Theorem]

Let [A | $\vec{b}$] be the augmented matrix of a system of m linear equations in n variables

a) The system is only consistent if and only if rank(A) = rank([A | $\vec{b}$])

b) If the system is consistent, then the number of parameters in general solution is the number of variables minus the rank of A

# of parameters = n - rank(A)

c) The system is consistent for all $\vec{b}\in {\mathbb{R}}^m$ if and only if rank(A) = m

A system of m linear equations in n variables is overdetermined if n < m, this is, if it has more equations than variables