Do vectors necessarily have full rank

linear algebravectors

My understanding is that an $m$ by $n$ matrix has full rank if and only if

  1. It has $\min\{m, n\}$ linearly independent columns, and
  2. 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

  1. It has 1 linearly independent column, and
  2. 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.

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