My understanding is that an $m$ by $n$ matrix has full rank if and only if
- It has $\min\{m, n\}$ linearly independent columns, and
- It has $\min\{m, n\}$ linearly independent rows
Now, a vector is a matrix with either one row or one column. That is, it is a matrix such that $\min\{m, n\} = 1$. It would thus seem that a vector has full rank if and only if
- It has 1 linearly independent column, and
- It has 1 linearly independent row
But that seems to hold trivially. So am I right in thinking that all vectors have full rank? And in particular, does the zero vector have full rank?
Best Answer
A single vector is a linearly independent set/family if and only if it is non-zero, as you can see from the definition of linear independence.