What Values of k Make These Vectors Linearly Dependent?

[Yankel]

I have 4 vectors:

(1,1,1,k), (1,1,k,1), (1,k,1,1), (k,1,1,1)

I need to find out for which values of k (if any) the vectors will be dependent.

I put them all to a matrix, and reduced it using the Gaussian method (homogeneous system). I received:

1st row: 1 1 1 k 0

2nd row: 0 (k-1) 0 (1-k) 0

3rd: 0 0 (k-1) (1-k) 0

4th: 0 0 0 (-2k+3-x^2) 0the reduction is correct, I verified with MAPLE.

I am stuck here, MAPLE say the solution is (0,0,0,0), meaning that the vectors are independent, but on the other hand, when I do it manually, I think is has a solution, for example, the last equation is:

(-2k+3-x^2)*x4=0

it can be either when x4 is 0 or when k is 1 or -3. I am confused...(Worried)

 

[alexmahone]

What are x and x4?

When k=1 all the vectors are equal, so that's a solution.
When k=-3 the sum of the vectors is the null vector, so that's also a solution.

 

[Yankel]

ah, sorry, confused x with k.

when k=1,-3 the system has infinite number of solutions, right ? so for these values the vectors are dependent...got it now, thanks !