What information do I get about the linear relations between a Matrix' lines from its rank?

Question

So, let's say I have a 4x2 matrix with a positive rank.

The maximum rank of the matrix is 2, and if the rank of the matrix is 1 I then know that one of the lines in the matrix is a scalar times the other line of the matrix.

Now, let's say I have a 6x3 matrix with a positive rank. The maximum rank of the matrix is 3, but I struggle to understand the difference between when the rank is 1 and when the rank is 2. I understand that a matrix' rank is the number of non-zero rows in its row echelon matrix, but I struggle to understand what it tells me.

Let's say I have rank 2. Does it mean that I get a linear combination of 2 lines that provides the 3rd? If so, what does rank 1 get you?

I really hope the question is coherent. Thank you and have a blessed day.

Answer 1

A 6x3 matrix has rank 3 if and only if all three rows (I would not say "lines") are inependent.

If it has rank 2 that means that two of the rows are independent and the third is a linear combination of those two.

If it has rank 1 that means that each row is a scalar multiple of one row.

Answer 2

If you think of a matrix as a linear transformation, then the rank is the dimension of the range. This is the definition that I find most illuminating, as well as most useful.

In the spirit of your question however, the rank is the maximum number of linearly independent rows (or columns). So you are correct in your example about the meaning of rank $2$. If the rank were $1$, it would mean that there is some row, such that each of the other rows is a scalar multiple of it. We can choose any nonzero row as the distinguished row. In general, if there rank is $r$ there are $r$ rows, such that every row of the matrix is a linear combination of those $r$ rows, and this isn't true for any number less than $r$.