Unit content
Linear independence
A collection of vectors is linearly independent when none of its vectors is redundant with respect to the others.
Formally, $\mathbf v_1,\ldots,\mathbf v_k$ are linearly independent when
$$c_1\mathbf v_1+\cdots+c_k\mathbf v_k=\mathbf0$$
forces
$$c_1=\cdots=c_k=0.$$
The all-zero choice is called the trivial combination.
Dependence means redundancy
If a nontrivial choice of coefficients gives the zero vector, the vectors are linearly dependent. Then at least one vector can be expressed as a linear combination of the others.
For example,
$$\mathbf v_1=(1,0),\qquad \mathbf v_2=(0,1),\qquad \mathbf v_3=(1,1)$$
are dependent because
$$\mathbf v_1+\mathbf v_2-\mathbf v_3=\mathbf0.$$
Equivalently,
$$\mathbf v_3=\mathbf v_1+\mathbf v_2.$$
Geometric intuition
In $\mathbb R^2$, two nonzero vectors are independent exactly when they are not parallel. In $\mathbb R^3$, three vectors are independent when none lies in the plane generated by the other two.
Independence asks whether each vector contributes a genuinely new direction to the collection.