# Full row-rank submatrix of full column-rank matrix over GF(2)?

#### xhumin

Let M be a m rows and n columns matrix over GF(2). And Let M' be a r rows and n columns submatrix of M (r <= m). Note that rows of M' is randomly selected from M. Is it possible that the M is full row-rank?

If it is, why?

If it is not, how about the case where r < m?

#### xhumin

Update:

1) I forgot to mention m > n.

2) I have run some experiments on Matlab. In experiments, I first randomly generated 10^6 matrices over GF(2). Each matrix has 288 rows and 216 columns. Turn out they all have the rank of 216. Then, I randomly generated 10^6 matrices over GF(2). Each matrix has 84 rows and 216 columns. Turn out they all have the rank of 84. So I am very confused now.

#### Idea

Let M be a m rows and n columns matrix over GF(2). And Let M' be a r rows and n columns submatrix of M (r <= m). Note that rows of M' is randomly selected from M. Is it possible that the M is full row-rank?

If it is, why?

If it is not, how about the case where r < m?
rank $$\displaystyle M$$= column rank $$\displaystyle M$$= row rank $$\displaystyle M$$$$\displaystyle \leq n$$ since $m>n$

what does this have to do with $M'$ ?