Matrix Rank

Definition:

The rank of a matrix $A$, denoted by $\mathrm{rank}(A)$ or $\rho (A)$, is the maximum number of linearly independent rows or columns in the matrix.

Intuitive Explanation

Think of the rank of a matrix as the number of "independent directions" or "degrees of freedom" in the matrix. In other words, it's the maximum number of rows or columns that are not a linear combination of the others.

Mathematical Definition

Let $A$ be an $m\times n$ matrix. The rank of $A$ is the maximum number of linearly independent rows or columns in $A$, denoted by: $\mathrm{rank}(A)=\max \{k:\text{there exists }k\text{ linearly independent rows or columns in A}\}$

Operator Rank

Let $\mathcal{A}\in \mathcal{L}(V)$ be a linear operator. The rank of $\mathcal{A}$ is the dimension of $\mathrm{im}(\mathcal{A})$, and, by Rank-Nullity Theorem, this also equals to $\mathrm{rank}(A)$, where $A$ is the matrix representation of $\mathcal{A}$ under any bases.

Properties

1. Row Rank = Column Rank: The number of linearly independent rows is equal to the number of linearly independent columns.
2. The rank of $A$ is equal to the rank of its transpose,$A^T$.
3. The rank of $A$ is less than or equal to the minimum of the number of rows ($m$) and the number of columns ($n$), i.e., $\mathrm{rank}(A)\le \min (m,n).$
4. $\dim (\ker \varphi )+\dim (\mathrm{im}\varphi )=\dim V$, from the Rank-Nullity Theorem