Explanation The vector equation in R⁴ given by c₁u₁+c₂u₂+c₃u₃=0 represents a linear system of 4 equations in 3 "unknowns". The fourth equation (from the fourth row in the coefficient matrix) is c₁0+c₂0+c₃(5)=0 so that c₃=0.

The third equation is c₁0+c₂(–1)+c₃0=0 so that c₂=0.

Finally, we can use the first equation which is c₁(4)+c₂(1)+c₃0=0 together with the previous knowledge that c₂=0 to see that c₁=0.

Therefore all the coefficients in the linear combination must be 0 and we have verified that the set {u₁, u₂, u₃} is linearly independent.