Vectors

Overview

Let $V$ be a vector space. Many important characterizations of $V$ involve lists of vectors in $V$ .

Spans

Let $V$ be a vector space over field $F$ . A linear combination of vectors $v_{1}, v_{2}, \dots, v_{n} \in V$ is a vector of the form

a_{1} v_{1} + a_{2} v_{2} + \dots + a_{n} v_{n}, where a_{1}, a_{2}, \dots, a_{n} \in F .

The set of all linear combinations of $v_{1}, v_{2}, \dots, v_{n}$ is called the span of $v_{1}, v_{2}, \dots, v_{n}$ , denoted $span (v_{1}, v_{2}, \dots, v_{n})$ . The span of the empty list of vectors is defined to be ${0}$ . We say $v_{1}, v_{2}, \dots, v_{n}$ spans $V$ if $span (v_{1}, v_{2}, \dots, v_{n}) = V$ .

The span of a list of vectors in $V$ is the smallest subspace of $V$ containing all the vectors in the list. In other words, the span is the subspace generated by the vectors.

Finite-Dimensional

A vector space is finite-dimensional if there exists a list of vectors in it that spans the space. Otherwise it is infinite-dimensional. A subspace of a finite-dimensional vector space is also finite-dimensional.

Linear Independence

Let $V$ be a vector space over field $F$ . A list $v_{1}, \dots, v_{n}$ of vectors in $V$ is called linearly independent if the only choice of $a_{1}, \dots, a_{n} \in F$ that makes $a_{1} v_{1} + \dots + a_{n} v_{n}$ equal $0$ is $a_{1} = \dots = a_{n} = 0$ . Otherwise the list is said to be linearly dependent. The empty list $⟨ ⟩$ is also declared to be linearly independent.

List $v_{1}, \dots, v_{n}$ is linearly independent if and only if each vector in $span (v_{1}, \dots, v_{n})$ has only one representation as a linear combination of $v_{1}, \dots, v_{n}$ .

Linear Dependence Lemma

Let $V$ be a vector space and $v_{1}, \dots, v_{n}$ be a linearly dependent list in $V$ . Then there exists $j \in {1, \dots, n}$ such that

$v_{j} \in span (v_{1}, \dots, v_{j - 1})$ , and
$span (v_{1}, \dots, v_{j - 1}, v_{j + 1}, \dots, v_{n}) = span (v_{1}, \dots, v_{n})$ .

Replacement Theorem

Let $V$ be a finite-dimensional vector space. Then the length of every linearly independent list of vectors in $V$ is less than or equal to the length of every spanning list of vectors.

Bases

Let $V$ be a vector space over a field $F$ . A basis of $V$ is a list of linearly independent vectors that span $V$ . Any two bases of a finite-dimensional vector space have the same length.

Criterion

Let $V$ be a vector space over field $F$ . A list $v_{1}, \dots, v_{n} \in V$ is a basis if and only if every $v \in V$ can be written uniquely in the following form where $a_{1}, \dots, a_{n} \in F$ :

v = a_{1} v_{1} + \dots + a_{n} v_{n} .

Linear Map Characterization

Let $V$ and $W$ be vector spaces over a field $F$ . Let $v_{1}, \dots, v_{n}$ be a basis of $V$ and $w_{1}, \dots, w_{n} \in W$ . Then there exists a unique linear map $T \in L (V, W)$ such that $T v_{j} = w_{j}$ for $j = 1, \dots, n$ . In particular,

T (a_{1} v_{1} + \dots + a_{n} v_{n}) = a_{1} w_{1} + \dots + a_{n} w_{n},

for coefficients $a_{1}, \dots, a_{n} \in F$ .