In M2x3(F), prove that the set?

Question

by columns:
{(1,0,0)(1,0,0), (0,1,0)(0,1,0), (0,0,1)(0,0,1), (1,1,1)(0,0,0), (0,0,0)(1,1,1)}
is linearly dependent.

Answer 1

Let a= [[1,0, 0], [1, 0, 0]], b= [[0, 1, 0], [0, 1, 0]], c=[[0, 0, 1], [0, 0, 1]], d=[[1, 1, 1], [0, 0, 0]], and e= [[0, 0, 0], [1, 1, 1]]. Then your set can be written as {a, b, c, d, e}. Since a+b+c-d-e = 0, this set of matrices is linearly dependent.