I have to questions for everyone,
if I carry out Gaussian elimination on a matrix of vectors that I am trying to work out if they are linearly independent, if I arrive at a row of zeros, are the vectors always linearly dependent?
with this set of vectors, I get a column of zeros however the answer says they are linearly independent (1,2,1,3), (2,0,-1,2), (1,1,1,2) (they are transposed)
Also, in what cases are a set of vectors linearly independent yet do not span Rn? Of course for example if you want to span R3 but only have 2 independent vectors, it is not possible, but are there any other instances?