[Math] is zero vector linearly dependent on other vectors

Question

If I have 3 column vectors which one of them is a zero vector, is it right to say they are linear dependent ? because (from my understanding) I can multiply out one of the other vector by scalar 0 and obtain the zero vector.Hence, the zero vector is not needed (redundant).

Thanks for your help!

Answer 1

Let's think about the definition of linear independence. It's not about redundancy; it's about whether a non-zero linear combination can equal the zero vector. Now if we have a set $\bf{v},\bf{w},\bf{0}$, there is a non-trivial linear combination that definitely works: $$0{\bf{v}}+0{\bf{w}}+1{\bf{0}}={\bf{0}}$$ That's why the set is linearly dependent.