Below you will find example sentences with "basis vectors". The examples show how this phrase is used in real sentences and which words often surround it.
Basis Vectors in a sentence
About this phrase
- Belongs to the word: basis
Example types with basis vectors
Below, the examples are grouped by length and sentence type:
Label the 8 basis vectors by the points of the projective plane over the field with 7 elements. (18 words)
An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. (19 words)
The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real ). (19 words)
Halmos 1974, p. 28, Ex. 9 Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them. (43 words)
Given a product of distinct orthogonal basis vectors of V, one can put them into standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation ). (41 words)
The set of all possible products of n orthogonal basis vectors with indices in increasing order, including 1 as the empty product forms a basis for the entire geometric algebra (an analogue of the PBW theorem ). (36 words)
Example sentences (14)
Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient.
Basic properties The following section uses the Cartesian coordinate system with basis vectors : and assumes that all vectors have the origin as a common base point.
An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other.
Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components.
Halmos 1974, p. 28, Ex. 9 Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them.
It is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis.
The dual basis vectors may be constructed as : where the denotes that the ith basis vector is omitted from the product.
The set of all possible products of n orthogonal basis vectors with indices in increasing order, including 1 as the empty product forms a basis for the entire geometric algebra (an analogue of the PBW theorem ).
A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space.
Given a product of distinct orthogonal basis vectors of V, one can put them into standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation ).
Label the 8 basis vectors by the points of the projective plane over the field with 7 elements.
The basis vectors used to represent the Jones vector need not represent linear polarization states (i.e. be real ).
The legacy of quaternionic analysis in vector analysis can be seen in the use of to indicate the basis vectors of R 3 : it is being thought of as the purely imaginary quaternions.
The vector A is still conventionally represented by a linear combination of basis vectors or a column matrix: : though the coordinates are now all complex-valued.