matrix linear independence

Question

the vectors:

v = [3, -10, 26]

u = [-3, 9, -28+k]

w = [1, -4, 10]

are linearly independent if k does not equal what???

Answer 1

Let A be the 3x3 matrix with u as the first column, v as the second, and w as the third.

{u, v, w} is a linearly independent set of vectors in lR³ only when det(A) ≠ 0.

Expanding along the first row,

det(A) = 3[90 + 4(k - 28)] + 3[-100 + 104] +1[-10(k - 28) - 234] = 2k - 8

So, the set is linearly independent as long as k ≠ 4.