Basis

30. Basis

Non-redundant set of vectors that span $ℝ^{𝑛}$

A basis of a vector space is a set of vectors that satisfies two conditions:

Linear Independence: no vector in the set can be written as a linear combination of the others — equivalently, the only solution to
$𝑐_{1} {\vec{𝑣}}_{1} + 𝑐_{2} {\vec{𝑣}}_{2} + \dots + 𝑐_{𝑛} {\vec{𝑣}}_{𝑛} = 𝟎$
is $𝑐_{1} = 𝑐_{2} = \dots = 𝑐_{𝑛} = 0$ .
Spanning: every vector $\vec{𝑣} \in 𝑉$ can be expressed as a linear combination of the basis vectors:
$\vec{𝑣} = 𝑐_{1} {\vec{𝑣}}_{1} + 𝑐_{2} {\vec{𝑣}}_{2} + \dots + 𝑐_{𝑛} {\vec{𝑣}}_{𝑛}$

Example

Consider the vector space $ℝ^{2}$ (the 2-dimensional Euclidean space). A common basis for $ℝ^{2}$ is ${𝑒_{1}, 𝑒_{2}}$ , where:

𝑒_{1} = [\begin{matrix} 1 \\ 0 \end{matrix}] 𝑒_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}]

This set is a basis because:

Linear Independence: The only solution to $𝑐_{1} 𝑒_{1} + 𝑐_{2} 𝑒_{2} = 𝟎$ is $𝑐_{1} = 𝑐_{2} = 0$ .
Spanning: Any vector $\vec{𝑣} = [\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}] \in ℝ^{2}$ can be written as $\vec{𝑣} = 𝑥 𝑒_{1} + 𝑥 𝑒_{2}$

This means ${𝑒_{1}, 𝑒_{2}}$ is a basis for $ℝ^{2}$ , and the dimension of $ℝ^{2}$ is 2.