Linear Algebra - Rank

Rank, Linearly Independent

The rank of a matrix A, $\mathrm{rank}(A)$ can be defined as
–the maximum number of linearly independent rows;
–or the maximum number of linearly independent columns;
–or the order of the largest non-zero minor of A.

Vectors $a1,a2,...,ak$ are linearly dependent if and only if there exist numbers c1, c2, … , ck not all zero, such that $c_1{a}_2+c_2{a}_2+...+c_k{a}_k=0$.
If $c_1=c_2=..c_k=0$, then they are linearly independent. if $c_1=1,c_2=..=c_k=0$, then they are still linear dependent according to the definition

$$\mathrm{rank}\left(\begin{array}{rrr}1& 3& 2\\ 2& 6& 4\\ -5& 7& 1\end{array}\right)=2$$

The first two rows are linearly dependent, therefore the maximum number of linearly independent rows is equal to 2.

Properties of the Rank

1. The column rank and the row rank of a matrix are equal.
2. $\text{rank}(AB)\le \min (\text{rank}(A),\text{rank}(B))$.
3. $\text{rank}(A)=\text{rank}(A{A}^T)=\text{rank}(A^{T}A)$.
4. Using the notion of rank, we can re-formulate the condition for non-singularity:
   5. If A is a square matrix of order n, then $\text{rank}(A)=n$​ ⇔ $\det (A)\ne 0$​ .