Linearly dependent and linearly independent vectors

Given a set of vectors we say that they are linearly dependent if one of these can be expressed as a linear combination of the others. In the plane, two vectors $$\vec{u}$$ and $$\vec{v}$$ that have the same angle are linearly dependent because it is true that $$\vec{v}=\lambda\vec{u}$$.

So, we can say that all the parallel vectors are linearly dependent upon one another since they all have the same angle.

If two vectors do not have the same angle, they are linearly independent since one of these vectors cannot be expressed as a linear combination of another.

In the plane three vectors are always linearly dependent because we can express one of them as a linear combination of the other two, as we previously commented.

Characteristics of linear independence:

1. Two vectors $$\vec{u}$$ and $$\vec{v}$$ are linearly independent if any linear combination of those equal to zero implies that the scalars $$\lambda$$ and $$\mu$$ are zero: $$$ \lambda\vec{u}+\mu\vec{v}=\vec{0} \Rightarrow \lambda=0 \ \text{ and } \ \mu=0$$$
2. Two vectors $$\vec{u}=(u_1,u_2)$$ and $$\vec{v}=(v_1,v_2)$$ are linearly independent if: $$$\dfrac{u_1}{v_1}\neq\dfrac{u_2}{v_2}$$$