Finite-Dimensional Vector Space

Prerequisite: Vector Space

Span

If $\mathbf{v}_1,\dots, \mathbf{v}_p\in \mathbb{R}^n$, then the set of all linear combinations of $\mathbf{v}_1, \dots, \mathbf{v}_p$ is denoted by $\text{Span}(\mathbf{v}_1, \dots, \mathbf{v}_p)$ and is called the subset of $\mathbb{R}^n$ spanned by $\mathbf{v}_1, \dots, \mathbf{v}_p$.

Linear Independence

An indexed set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_p\}$ in $\mathbb{R}^n$ is said to be linearly dependent if

$$x_1\mathbf{v}_1 + \dots + x_p \mathbf{v}_p = \mathbf{0}$$

has only the trivial solution. In other words, they are call linearly dependent if there exists non-zero solution.

Proposition: If $p>n$, then the set is linearly dependent.