Rules about linear independence in matrices

Question

is there a rule about the maximum number of possible linearly independent columns allowed in a matrix?

For example, can a 4x6 matrix have 5 linearly independent columns?

and does the same concept apply for rows?

Answer 1

The maximum number of LI rows or columns possible for an $n\times m$ matrix would be $\min\{n,m\}$.

In your example, each of your columns is a vector in $\Bbb R^4$, and there can only be a maximum of $4$ LI vectors in $\Bbb R^4$.

Answer 2

That is a wonderful question!

Let's say we're talking about matrices with "n" rows and "m" columns.

Then the maximum number of linearly independent rows is min(n, m). The maximum number of linearly independent columns is ALSO min(n, m). In fact, the most fundamental theorem of linear algebra says that the number of linearly independent rows must equal the number of linearly independent columns (regardless of the shape of the matrix). This number is called the rank of the matrix.

Sometimes you will hear people say "row rank" or "column rank"- these terms are very helpful in my opinion (they still confuse me now, and I'm a PhD student in applied mathematics who works constantly with linear algebra).