Posted by Axel on .
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)

Math 
MathMate,
The vectors form a linearly dependent set in ℜ^{3} 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.