Editing Rank (linear algebra) (section)

== Examples ==
The matrix
<math display="block">\begin{bmatrix}
1 & 0 & 1\\
0 & 1 & 1\\
0 & 1 & 1
\end{bmatrix}</math>
has rank 2: the first two columns are [[linear dependence|linearly independent]], so the rank is at least 2, but since the third is a linear combination of the first two (the first column plus the second), the three columns are linearly dependent so the rank must be less than 3.

The matrix
<math display="block">A=\begin{bmatrix}1&1&0&2\\-1&-1&0&-2\end{bmatrix}</math>
has rank 1: there are nonzero columns, so the rank is positive, but any pair of columns is linearly dependent.  Similarly, the [[transpose]]
<math display="block">A^{\mathrm T} = \begin{bmatrix}1&-1\\1&-1\\0&0\\2&-2\end{bmatrix}</math>
of {{mvar|A}} has rank 1.  Indeed, since the column vectors of {{mvar|A}} are the row vectors of the [[transpose]] of {{mvar|A}}, the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e., {{math|1=rank(''A'') = rank(''A''<sup>T</sup>)}}.