Span & Basis

For all the definitions below, consider $B$ to be a non-empty subset of the vector space $V\;\text{over}\;F$.

Linear Combination

For a non-empty finite subset of $V$ (say $B$), a vector $x \in V$ is a linear combination of vectors in $B$ iff

Where $x_k \in B$, $a_k \in F$ and $n$ is the size of $B$.

All linear combinations of a vector space, is an element of itself.

Span

For a non-empty finite subset of $V$ (say $B$), the set of all possible linear combinations of vectors in $B$. Denoted as $\text{Span}\;B$. Always a subspace of $V$. Obviously $B \subseteq \text{Span}\;B$.

As $B \subseteq V$ and $V$ is closed over vector addition and scalar multiplication, $\text{Span}\;B \subseteq V$.

If $\text{Span}\;B = V$ then it is read as $B\;\text{spans}\;V$.

Span is a subspace

Smallest subspace

$\text{Span}\;B$ is the smallest subspace of $V$, containing $B$.

Linear Independence

$B$ is linear independent on $V\;\text{over}\;F$ iff:

Empty sets are linearly independent by definition. Otherwise $B$ is linearly dependent on $V\;\text{over}\;F$.

If $B$ contains $\underline{0}$, it is linearly dependent. Because coefficient of $\underline{0}$ can be non-zero and still produce a zero sum.

If $B$ is a singleton set $\{x\}$, it is linearly independent iff $x \neq \underline{0}$.

Any subset of a linear independent set is also linear independent. Any superset of a linear dependent set is also linear dependent.

Theorem 1

If $B$ is linearly dependent then at least one vector in $B$ can be expressed as a linear combination of other vectors in $B$.

Theorem 2

If $B$ is linearly independent and $x \not\in \text{Span}\;B$ then $B \cup \set{x}$ is linearly independent.

Basis

$B$ is a basis of $V\;\text{over}\;F$ iff:

• $B$ is linearly independent
• $B$ spans $V$

Suppose a basis of $V$ has $n$ elements. Any subset of $V$ with:

• more than $n$ vectors is linearly dependent
• fewer than $n$ vectors cannot span $V$

If a set spans but no proper subset spans, it is a basis. Any linearly independent set can be extended to a basis.

Any element in a vector space can be uniquely expressed as a linear combination of its basis vectors.

Every spanning non-empty subset of a vector space contains a basis.

If $B$ is a finite, non-empty subset of $V$ and linearly independent, and $x \not\in \text{Span}\;B$, then $B\cup\set{x}$ is linearly independent.

Any linearly independent subset $B$ of $V$ can be extended to a basis.

If $B$ is maximally linearly independent (no superset of $B$ is linearly independent), it is a basis. If $B$ is minimally spanning, it is a basis.

Hamel Basis

A non-finite basis of $V$ over $F$ where only finite linear combinations are allowed. Every vector space has a Hamel basis.

Dimension

Number of elements in the basis of a vector space $V\;\text{over}\;F$. Denoted as $\dim V$. Constant for all bases of $V$ (obvious from above theorem).

If $\dim V = n$ then any linearly independent set of $n$ vectors in $V$ is a basis of $V$. If $\dim V = n$ then any spanning set of $n$ vectors in $V$ is a basis of $V$.