Frames and bases for finite a dimension vector space

If $v_1,\ldots,v_n$ are some elements of a vector space ${\cal{V}}$ , by a linear combination of $v_1,\ldots,v_n$ we mean an element in ${\cal{V}}$ of the form $a_1 v_1 + \cdots +a_n v_n$ , with $a_i \in \mathbb{F}, i=1,\ldots,n$ .

Let be a subset of element of ${\cal{V}}$ . The set of all linear combinations of elements of is called the span of and is denoted by ${\mbox{\rm {span}}} S$ .

A subset $S=\{v_i\}_{i=1}^n$ of ${\cal{V}}$ is said to be linearly independent if and only if

$\displaystyle a_1 v_1 + \cdots +a_n v_n= 0,\quad \implies \quad a_i=0, i=1,\ldots,n.$

A subset is said to be linearly dependent if it is not linearly independent.

${\cal{S}}$ is said to be a basis of ${\cal{V}}$ if it is linearly independent and ${\mbox{\rm {span}}} S={\cal{V}}$ . The dimension of a finite dimensional vector space ${\cal{V}}$ is the number of elements in a basis for ${\cal{V}}$ . The number of elements in a set is termed the cardinality of the set.

If the number of elements spanning a finite dimensional vector space ${\cal{V}}$ is larger than the number of elements of a basis for the same space, the set is called a redundant frame. In other words a redundant frame (henceforth called simply a frame) for a finite dimensional vector space ${\cal{V}}$ is a linearly dependent set of vectors spanning ${\cal{V}}$ .

Let $\{v_i\}_{i=1}^n$ be a spanning set for ${\cal{V}}$ . Then every $v \in {\cal{V}}$ can be expressed as

$\displaystyle v= a_1 v_1 + \cdots, a_n v_n,$ with $\displaystyle ,\quad a_i \in \mathbb{F}, i=1,\ldots,n.$

If the spanning set $\{v_i\}_{i=1}^n$ is a basis for ${\cal{V}}$ the numbers $a_i \in \mathbb{F}, i=1,\ldots,n$ in the above decomposition are unique. In the case of a redundant frame, however, these numbers are not longer unique. For further discussion about redundant frames in finite dimension see [1].

Highly Nonlinear Approximations for Sparse Signal Representation

Frames and bases for finite a dimension vector space