Linear Combinations and Span

In $xy$ plane, $\hat{i}$ and $\hat{j}$ are called basis vectors.

Using a linear combination (scaled sum) of $\hat{i}$ and $\hat{j}$, we can reach any vector in 2D space. In this case, the 2D space is the span (set of all possible linear combinations) of $\hat{i}$ and $\hat{j}$.

Notice that the span is not the same for basis vectors that line up or are 0.

The property that describes a basis vector that adds another dimension to span is linear independence.

A more formal definition for linear independence within a set of vectors is ${\vec{v}}_1,{\vec{v}}_2,\dots {\vec{v}}_p$ are linearly independent if and only if

$$c_1{\vec{v}}_1+c_2{\vec{v}}_2\cdots =0$$

has only trivial solution $\vec{c}=\vec{0}$

A set of vectors is linearly dependent if one of the following happen:

1. $v_i=\vec{0}$
2. ${\vec{v}}_i=k{\vec{v}}_m$
3. A vector is a linear combination of the rest
4. The number of vectors is greater than the number of components of each vector.

Technical definition of basis: The basis of a vector space is a set of linearly independent vectors that span the full space.