The vectors V 1 2 4 u 2 2 11 k and w 3 6 0 are linearly

The vectors V = [1 2 -4], u = [2 -2 11 + k], and w = [-3 6 0] are linearly independent if and only if k not equal .

Solution

system of linear equation form given vector:

v+2u = -3, 2v-2u=6

solve the system :

we get v = 1 and u = -2

then third equation : -4v + (11+k)u = 0

put the value of v and u:

-4*1 + (11+k)*-2 = 0

- 4 - 22 - 2k = 0

-26 - 2k = 0

k = -26/2 = -13

thus v, u and w are linearly independent if and only if k is not equal to -13.