# Linear dependency among columns and rows

Singular matrix is defined as square matrix with the determinant of zero. The determinant of zero occurs when matrix columns are linearly dependent (i.e. one of the columns can be defined as a linear combination of other columns).

However, some sources also note that the determinant can be zero when there is linear dependency not only among the columns but also among the rows of a square matrix. Therefore, I wanted to ask:

Does linear dependency among columns imply that there is automatically linear dependency among rows of data?

Note: this is a follow-up from an older thread: Singular Matrix and Linear Dependency

• This answer is perhaps too limited to be of much use or insightful. Standard theorems about dimensions of vector spaces provide clear, simple relationships between the numbers of rows, columns, and dimension of the null space of any matrix. A good example to ponder is the $1\times 2$ matrix $\pmatrix{1&1},$ which has an obvious linear dependency among its columns and just as obviously has no linear dependency among its rows.