Rank of an Outer Product

Every column of $u{v}^T$ is a scalar multiple of $u$.

The column space is therefore spanned by a single nonzero vector.

1. 1. Examine the columns

   The $j$-th column of $u{v}^T$ equals $v_{j}u$, so every column is a scalar multiple of the single vector $u$.

2. 2. Conclude the rank

   Since $u$ is nonzero and all columns lie in the one-dimensional span of $u$, the rank of $u{v}^T$ is $1$.

Answer: 1