Rank

Definition

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

Intuition

The rank of a matrix is the number of dimensions the column space can span.

If it's easier, the use of "column space" can be taken to mean each vector in the matrix.

Examples

Rank:

Rank: because there's 1 in the rightmost column

System-Rank Theorem

Theorem

Let be the augmented matrix of a system of linear equations in variables

The system is consistent if and only if
If the system is consistent, then the number of parameters in the general solution is the number of variables minus the rank of

The system is consistent for all if and only if

System Rank Theorem

, the system is inconsistent
The system-rank theorem does not apply.
, so the system will not be consistent for every

System Rank Theorem with Variables

If :
- If :
  - , so the system is consistent
  - parameter, so there are infinitely many solutions
- If :
  - , so the system is inconsistent with no solutions
If :
- , so the system is consistent
- parameters, so there is a unique solution

For the constant matrix , is it required that for the system to be consistent

Underdetermined

Definition

A linear system is underdetermined if there are fewer equations than variables

A consistent underdetermined system has infinitely many solutions

Overdetermined

Definition

A linear system is overdetermined if there are more equations than variables

Overdetermined systems are often inconsistent

Full Rank

Definition

A matrix is full rank when its rank equals the largest possible for a matrix with its dimensions, which is .

Rank Deficiency

Definition

A matrix is said to be rank-deficient when it is not full rank. The rank deficiency of a matrix is the difference between the maximum rank possible, and the rank of the matrix.