The 3 vectors $u=(1,2,3)$, $v=(2,5,7)$, $w=(1,3,5)$ are linearly independent. But they should be linearly dependent as they lie on the same plane, then where this concept goes on such type of questions?
[Math] Three vectors in a plane are linearly dependent
linear algebra
Best Answer
It’s true that any set of three points is coplanar, but for linear dependence, we only care about planes that pass through the origin, as those are the two-dimensional vector subspaces of $\mathbb R^3$. Those planes are the set of all linear combinations of a pair of linearly-independent vectors, whereas planes that don’t include the origin can’t be generated that way. So, in the context of linear algebra, the qualifier “through the origin” is often understood and omitted to save space.