Question

For which real values of x do the following vectors form a linearly dependent set in R3?

v1= (x, -1/9, -1/9)
v2= (-1/9, x, -1/9)
v3= (-1/9, -1/9, x)

Answer 1

The vectors form a linearly dependent set in ℜ³ if the determinant of the matrix formed by the three vectors is zero.

Evaluate the determinant:
|x,k,k|
|k,x,k|
|k,k,x|
in terms of x and k, where k can eventually substituted for -1/9.
Solve the resulting cubic equation for x in terms of k. It actually factors nicely into (x+?)(x-??)².

In fact, you should be able to determine ?? by inspection.

Post if you need more help.