Linearly Independent Sets and Spanning Sets in Linear Spaces

# Linearly Independent Sets and Spanning Sets in Linear Spaces

## Linearly Independent Sets

 Definition: Let $X$ be a linear space over $\mathbb{R}$ (or $\mathbb{C}$). A subset $\{ x_1, x_2, ..., x_n \} \subseteq X$ is said to be Linearly Independent in $X$ if the equation $a_1x_1 + a_2x_2 + ... + a_nx_n = 0$ implies that $a_1 = a_2 = ... = a_n = 0$. The subset $\{ x_1, x_2, ..., x_n \}$ is said to be Linearly Dependent if it is not linearly independent, that is, there exists $a_1, a_2, ..., a_n \in \mathbb{R}$ (or $\mathbb{C}$) that are not all zero such that $a_1x_1 + a_2x_2 + ... + a_nx_n = 0$.

For example, consider the linear space $\mathbb{R}^2$. Then for all $s, t \in \mathbb{R}$ with $s, t \neq 0$, the set $\{ (s, 0), (0, t) \} \subseteq \mathbb{R}^2$ is a linearly independent set since if $a, b \in \mathbb{R}$ and:

(1)
\begin{align} \quad a(s, 0) + b(0, t) = (as, 0) + (0, bt) = (as, bt) = (0, 0) \end{align}

Implies that $as = 0$ and $bt = 0$. But since $s, t \neq 0$ we have that $a = 0$ and $b = 0$.

It can easily be shown that for the vector space $\mathbb{R}^n$, every subset of $X$ of size greater than $n$ must be linearly dependent.

## Spanning Sets

 Definition: Let $X$ be a linear space over $\mathbb{R}$ (or $\mathbb{C}$). For $x_1, x_2, ..., x_n \in X$, the Span of $x_1, x_2, ..., x_n$ denoted $\mathrm{span} (x_1, x_2, ..., x_n)$ is the set of all linear combinations of the elements $x_1, x_2, ..., x_n$. A subset $\{ x_1, x_2, ..., x_n \} \subseteq X$ is said to be a Spanning Set of $X$ if $\mathrm{span} (x_1, x_2, ..., x_n) = X$.

Equivalently, a subset $\{ x_1, x_2, ..., x_n \} \subseteq X$ is said to be linearly independent in $X$ if the only way a linear combination of the elements $x_1, x_2, ..., x_n$ is equal to $0$ is by setting the coefficients of the linear combination all to $0$.

It can easily be shown that for the vector space $\mathbb{R}^n$, every subset of $X$ of size less than $n$ cannot span $\mathbb{R}^n$.