Bases

Definition: Basis

A basis of a vector space $(V,F,+,\cdot )$ is a linearly independent spanning set for it.

Equivalent Definition

A set of vectors is a basis for a vector space $(V,F,+,\cdot )$ if and only if it is a maximal linearly independent set.

PROOF

TODO

Equivalent Definition

A set of vectors is a basis for a vector space $(V,F,+,\cdot )$ if and only if it is a minimal spanning set.

PROOF

TODO

Dimension

Theorem: Number of Basis Elements

All bases of a given vector space have the same cardinality.

PROOF

TODO

Definition: Dimension of a Vector

The dimension of a vector space $(V,F,+,\cdot )$ is the cardinality of its bases.

NOTATION

$$\dim (V)$$

Properties

Theorem: Basis Representation

Let $(V,F,+,\cdot )$ a vector space.

If $B$ is a basis of $(V,F,+,\cdot )$, then each vector of $(V,F,+,\cdot )$ can be uniquely expressed as a linear combination of the elements of $B$.

Note: Uniqueness

Here, “uniquely” means that no two vectors are expressed as a linear combination which has the exact same coefficients in front of the exact same basis vectors.

PROOF

TODO

Theorem: Basis Criterion

Let $(V,F,+,\cdot )$ be a vector space.

Every set $B$ of linearly independent vectors whose cardinality is equal to the dimension $\dim (V)$ is a basis for $(V,F,+,\cdot )$.

PROOF

TODO

Theorem: Steinitz Exchange Lemma

Let $B=\{{\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n\}$ be a basis of a finitely generated vector space $(V,F,+,\cdot )$.

In each set $\{{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m\}\subset V$ of $m$ linearly independent vectors there are $n-m$ vectors in $B$ (without loss of generality ${\mathbf{v}}_{m+1},\cdots ,{\mathbf{v}}_n$) such that

$$\{{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m,{\mathbf{v}}_{m+1},\cdots ,{\mathbf{v}}_n\}$$

is also a basis of $(V,K,+,\cdot )$.

TIP

This means that for every set $\{{\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m\}\subset V$ of $m$ linearly independent vectors, we can find $m$ vectors in $B$ which we can replace with ${\mathbf{u}}_1,\cdots ,{\mathbf{u}}_m$ and still obtain a basis of $(V,K,+,\cdot )$.

PROOF

TODO

Ordered Bases

Definition: Ordered Basis

If $B=\{{\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n\}$ is a basis of an $n$-dimensional vector space $(V,F,+,\cdot )$, then any $n$-tuple of the elements of $B$ is called an ordered basis of $(V,F,+,\cdot )$.

Definition: Coordinate Vector

Let $B=({\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n)$ be an ordered basis of an $n$-dimensional vector space $(V,F,+,\cdot )$.

If $\mathbf{v}\in V$ has the basis representation $\mathbf{v}=v_1{\mathbf{b}}_1+\cdots +v_n{\mathbf{b}}_n$, then the coordinate vector of $\mathbf{v}$ with respect to the basis $B$ is the column vector

$${}_B\mathbf{v}\stackrel{\text{def}}{=}\left[\begin{array}{c}{v}_1\\ ⋮\\ v_n\end{array}\right]\in F^n.$$