[jay_shark]

Think of A as the matrix and X as a single column vector.

ie , X={x1,x2,x3,...xn }T where T is the transpose .

And A can be written as {V1 V2 V3 ... Vn} and the Vi's are column vectors .

In other words , you wish to solve for x1*V1 + x2*V2 +...xn*Vn = ( 0,0,0)

The rhs is the 0 vector which may be written as (0,0,0) .

ie , V1=(0,0,1) V2= (1,0,1) V3= (0,1,1) and the xi's are scalars .

So we wish to solve for x1*(0,0,1) + x2*(1,0,1) + x3*(0,1,1) =(0,0,0)

Or x2=0 , x3=0 and x1+x2+x3=0 which implies x1=0 .

Since all three of the xi's are zero , this implies that the vectors V1,V2 and V3 are linearly independent .

Also it's important to know that a set of vectors is linear dependent if and only if some vi can be written as a linear combination of the others .If you can prove independence , then you can prove that a single vector cannot be written as a linear combination of the others .