> For the complete documentation index, see [llms.txt](https://muhans-notebook.gitbook.io/computational-optimization/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://muhans-notebook.gitbook.io/computational-optimization/basics.md).

# Basics

### Rank

*The rank of a matrix is the maximum number of its* [*linearly independent column*](https://www.sciencedirect.com/topics/mathematics/linearly-independent-column) *vectors (or row vectors).*

It also can be shown that the columns (rows) of a [square matrix](https://www.sciencedirect.com/topics/mathematics/square-matrix) are linearly independent only if the matrix is nonsingular. In other words, the rank of any [nonsingular matrix](https://www.sciencedirect.com/topics/mathematics/nonsingular-matrix) of order *n* is *n.*

### Nonsingular Matrix

An $$n \times n$$ matrix $$A$$ is called *nonsingular* or *invertible matrix* if there exists an $$n \times n$$ matrix $$B$$ such that

$$
AB=BA=I.
$$

If $$A$$ does not have an inverse, $$A$$ is called *singular matrix*.

### **Orthogonal Matrices**

**Definition.** A matrix $$A$$ is **orthogonal** if $$A^TA=AA^T=I$$.

### Range

The [**range**](https://mathworld.wolfram.com/ColumnSpace.html) (also called the [column space](https://mathworld.wolfram.com/ColumnSpace.html) or image) of a $$m \times n$$ matrix $$A$$ is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://muhans-notebook.gitbook.io/computational-optimization/basics.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.