Linear independence

Definition

A subset $S$ of a vector space $V$ is called linearly dependent if there exist a finite number of distinct vectors ${\mathbf{u}}_1,\cdots ,{\mathbf{u}}_n$ in $S$ and scalars $a_1,\cdots ,a_n$, not all zero, such that

$\sum \_{i=1}^{n}a\_i\mathbf{u}\_i=0.$

In this case we also say that the vectors of $S$ are linearly dependent.

A subset $S$ of a vector space $V$ that is not linearly dependent is called linearly independent. We also say that the vectors of $S$ are linearly independent.

Theorems

Theorem 1

Let $V$ be a vector space over a field $F$, and let $S_1\subseteq S_2\subseteq V$.

If $S_1$ is linearly dependent, then $S_2$ is also linearly dependent.

Proof

Since $S\_1$ is linearly dependent, we can choose some vectors $\mathbf{v}\_1,\cdots ,\mathbf{v}\_n$ such that for some scalars $a\_1,\cdots ,a\_n$,

$$\sum \_{i=1}^{n}a\_i\mathbf{v}\_i=0.$$

Then, since $S\_1\subseteq S\_2$, we know that $\check{v}\_i\in S\_2$ for all $i=1,\cdots ,n$. Hence by definition the vectors of $S\_2$ are linearly dependent. \tag*{$||$}

Corollary

Let $V$ be a vector space over a field $F$, and let $S_1\subseteq S_2\subseteq V$.

If $S_2$ is linearly independent, then $S_1$ is also linearly independent.

Theorem 2

Let $S$ be a linearly independent subset of a vector space $V$, and let $\mathbf{v}$ be a vector in $V$ that is not in $S$.

Then $S\cup \{\mathbf{v}\}$ is linearly dependent if and only if $\mathbf{v}\in \text{span}(S)$.