##  [Rank](/rank) 

  ##  [Rank](https://natural.quantumdictionary.io/rank-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The rank of a matrix or linear transformation is the dimension of its image (column space), equivalently the maximum number of linearly independent columns or rows. It measures how many independent directions the map sends input vectors to.

 

 

 

 

 





 

 



 ##  [Rank](https://mathlogic.quantumdictionary.io/rank-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The rank of a linear map (or matrix) is the dimension of its image (column space); equivalently, the maximum number of linearly independent columns or rows in any matrix representing the map.

 

 

 

 

 





 

 



 ##  [Rank](https://puremath.quantumdictionary.io/rank-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A nonnegative integer (or cardinal) describing the maximal number of linearly independent rows or columns of a matrix, equivalently the dimension of the image (column space) of a linear map; it quantifies the effective linear degrees of freedom of the operator or matrix.

 

 

 

 

 





 

 



 ##  [Rank](https://algebra.quantumdictionary.io/rank-3) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

Rank is the maximum number of linearly independent elements in a set (over a specified field or domain) or, equivalently in linear algebra, the dimension of the image (column space or row space) of a linear map or matrix, expressed as an integer.