What you wrot eis correct, but you're trying too hard. The problem does not require an extensive construction. For example:
satisfies the required conditions.
Problem:
Find three vectors in R3 which are linearly dependent, and are such that any two of them are linearly independent.
What I have so far:
{v1, v2, v3} is a linearly dependent set.
{v1, v2} is a linearly independent set.
{v2, v3} is a linearly independent set.
{v1, v3} is a linearly independent set.
a*v1 + b*v2 + c*v3 = 0, where not all a, b, and c = 0 due to linearly dependence.
If a is not 0 then, v1 = - [ ( b*v2 + c*v3 ) / a ]
If b is not 0 then, v2 = - [ ( a*v1 + c*v3 ) / b ]
If c is not 0 then, v3 = - [ ( a*v1 + b*v2 ) / c ]
And, I am stuck. I don't know what to do after this, much less use this to find three vectors in R3 that satisfy this.
Thank you in advance,
-Z