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A set of linearly independent vectors taken in a definite order and placed at a common origin. Any three non-parallel vectors not lying in one plane can serve as a frame for the vectors in space. If the vectors building the frame are mutually orthogonal, then the frame is called orthogonal; if in this case the length of the vectors is equal to one, the frame is called orthonormal.

====Comments====
Usually a frame is called a [[Basis|basis]] (of vectors in space). In this sense, the word  "frame"  is also used in physics (frame of reference, cf. [[Reference system|Reference system]]). For Frénet frame see [[Frénet trihedron|Frénet trihedron]].

A framing of an  $  n $-
dimensional differentiable manifold  $  M $
is a vector bundle isomorphism of its tangent bundle  $  TM $
with the trivial bundle  $  M \times \mathbf R  ^ {n} $(
so that  $  M $
is parallelizable). Using the standard basis  $  ( e _ {1} \dots e _ {n} ) $
of  $  \mathbf R  ^ {n} $
such an isomorphism defines a frame field: it assigns to every  $  x \in M $
a frame, or basis, of the tangent space at that point.

The frame bundle over a manifold  $  M $
is the [[Principal fibre bundle|principal fibre bundle]] with structure group  $   \mathop{\rm GL} _ {n} ( \mathbf R ) $
whose fibre over  $  x \in M $
is the collection of all bases (frames) of  $  T _ {x} M $, 
the tangent space at that point.

A  $  k $-
frame  $  v  ^ {k} $
in  $  \mathbf R  ^ {n} $
is an ordered set of  $  k $
linearly independent vectors. Let  $  V _ {n,k} $
denote the set of all  $  k $-
frames in  $  \mathbf R  ^ {n} $. 
Let  $  G ( k) $
be the subgroup of  $   \mathop{\rm GL} _ {n} ( \mathbf R ) $
leaving a fixed frame  $  v _ {0}  ^ {k} $
invariant. Then  $  V _ {n,k} =  \mathop{\rm GL} _ {n} ( \mathbf R ) / G ( k) $. 
Thus,  $  V _ {n,k} $
has a real-analytic structure. It is called the Stiefel manifold of  $  k $-
frames in  $  n $-
space.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>