The vectors u=[-4 -8 -28], v=[-2 -6 -12+k], w=[3 12 33] are linearly independent if and only if k does not equal...

Question

Hi there, so need to find the value of k here. i am really stuck. thanks.

Answer 1

au +bv + cw = 0 iff a = b = c = 0

-4a - 2b + 3c = 0 (1)

-8a - 6b + 12c = 0 (2) (2) - 2(1): -2b + 6c = 0

-28a + (-12 + k)b + 33c = 0 (3) (3) - 7(1): (2 + k)b + 12c = 0

To have only zero solution a = b = c = 0

determinant of last system must be not equel zero zero: - 24 - 12 - 6k ≠ 0; - 6k ≠ 36; k ≠ -6