has full column rank means that the columns of are linearly independent. see that if are the columns of and then

so if then and thus for all because are linearly independent. hence the only solution of is

i think by you mean the transpose of assuming that the entries of come from a field the matrix has colums and these columns are in we know that

for the latter case,, thus we can only have y'y >= 0because A has more rows then columns, then there is an X such that A'x = 0

What I dont understand is the the bold and underline part.

P/S: this question comes from pg 835, of William Greene Econometric Analysis 5th edition textbook.

Thanks in Advance !!!

thus if then any vectors in are linearly dependent. now has columns, say and so they are linearly dependent, i.e. there exists

such that this also can be written as: