Vector space/Linear independence/Also infinite/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. A family of vectors ${}v_{i}$, ${}i\in I$, (where ${}I$ denotes a finite index set) is called linearly independent if an equation of the form

$\sum _{i\in I}s_{i}v_{i}=0{\text{ with }}s_{i}\in K$

is only possible when ${}s_{i}=0$

for all ${}i$.

If a family is not linearly independent, then it is called linearly dependent. A linear combination ${}\sum _{i\in I}s_{i}v_{i}=0$ is called a representation of the null vector. It is called the trivial representation if all coefficients ${}s_{i}$ equal ${}0$ and, if at least one coefficient is not ${}0$, a nontrivial representation of the null vector. A family of vectors is linearly independent if and only if one can represent with the family the null vector only in the trivial way. This is equivalent with the property that no vector of the family can be expressed as a linear combination by the others.

Example

The standard vectors in ${}K^{n}$ are linearly independent. A representation

${}\sum _{i=1}^{n}s_{i}e_{i}=0\,$

just means

${}s_{1}{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}+s_{2}{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}+\cdots +s_{n}{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}\,.$

The ${}i$-th row yields directly ${}s_{i}=0$.

Example

The three vectors

${\begin{pmatrix}3\\3\\3\end{pmatrix}},\,\,{\begin{pmatrix}0\\4\\5\end{pmatrix}},\,{\text{ and }}\,{\begin{pmatrix}4\\8\\9\end{pmatrix}}$

are linearly dependent. The equation

${}4{\begin{pmatrix}3\\3\\3\end{pmatrix}}+3{\begin{pmatrix}0\\4\\5\end{pmatrix}}-3{\begin{pmatrix}4\\8\\9\end{pmatrix}}={\begin{pmatrix}0\\0\\0\end{pmatrix}}\,$

is a nontrivial representation of the null vector.

Remark

The vectors ${}v_{1}={\begin{pmatrix}a_{11}\\\vdots \\a_{m1}\end{pmatrix}},\ldots ,v_{n}={\begin{pmatrix}a_{1n}\\\vdots \\a_{mn}\end{pmatrix}}\in K^{m}$ are linearly dependent if and only if the homogeneous linear system

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}$

has a nontrivial solution.

For an infinite family of vectors, there is also a concept of linear independence.

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Then a family of vectors ${}v_{i}$, ${}i\in I$, in ${}V$ is called linearly independent if an equation

$\sum _{i\in J}s_{i}v_{i}=0{\text{ with }}s_{i}\in K{\text{ for a finite subset }}J\subseteq I$

is only possible if ${}s_{i}=0$

for all ${}i$.

By this definition, linear independence of an arbitrary family is reduced to the finite case. Keep in mind that in a vector space, there are no infinite sums. An expression like ${}\sum _{n\in \mathbb {N} }s_{n}v_{n}=0$ does not have any sense.

Lemma

Let ${}K$ be a field, let ${}V$ be a ${}K$-vector space, and let ${}v_{i}$, ${}i\in I$,

be a family of vectors in ${}V$. Then the following statements hold.

1. If the family is linearly independent, then for each subset ${}J\subseteq I$, also the family ${}v_{i}$ , ${}i\in J$, is linearly independent.
2. The empty family is linearly independent.
3. If the family contains the null vector, then it is not linearly independent.
4. If a vector appears several times in the family, then the family is not linearly independent.
5. A single vector ${}v$ is linearly independent if and only if ${}v\neq 0$.
6. Two vectors ${}v$ and ${}u$ are linearly independent if and only if ${}u$ is not a scalar multiple of ${}v$ and vice versa.

Proof

See exercise.

$\Box$