Do vectors necessarily have full rank?

Question

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?

Answer 1

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.

Answer 2

The rank of a matrix is the dimension of the vector space spanned by its columns (see Wikipedia: https://en.wikipedia.org/wiki/Rank_(linear_algebra)).

Since a vector is a 1-column matrix, it will therefore have "full rank" in the matrix sense if the space it spans is of dimension 1 (same as the number of columns)

If the vector is non-zero, this is obviously true. If the vector is zero (all its elements are 0), then the space it spans is reduced to itself only (multiplying the zero vector with any number always gives back the zero vector). So in this case only, it will not have full rank in the matrix sense.

The vector