Below you will find example sentences with "basis vectors". The examples show how this phrase is used in natural context and which words often surround it.

Basis Vectors in a sentence

Corpus data

  • Displayed example sentences: 14
  • Discovered as a combination around: basis
  • Corpus frequency in the collocation scan: 8
  • Phrase length: 2 words
  • Average sentence length: 28 words

Words in this phrase

basis vectors

Meanings of words in this phrase

  • basis can mean: A physical base or foundation. Frequent word
  • vectors can mean: plural of vector Less frequent word

Sentence profile

  • Phrase position: 6 start, 8 middle, 0 end
  • Sentence types: 14 statements, 0 questions, 0 exclamations

Usage patterns

These patterns show short text windows around the phrase in the example sentences:

combination of basis vectors or a 1× combination of basis vectors whose coefficients 1× distinct orthogonal basis vectors of v 1× indicate the basis vectors of r 1× list the basis vectors in a 1× n orthogonal basis vectors with indices 1× non fixed basis vectors that change 1× of orthogonal basis vectors is often 1×

Sentence openings and endings

Common sentence openings:

a vector can 1× an orthonormal basis 1× basic properties the 1× choosing a set 1× given a basis 1× given a product 1×

Common sentence endings:

all complex valued 1× combination of them 1× common base point 1× coordinates or components 1× e be real 1× from the product 1×

Context words

These words often appear near this phrase in the example sentences:

vector 7× linear 6× orthogonal 4× set 3× combination 3× order 3× product 3× considering 2× convenient 2× given 2× space 2× element 2×

Related phrases

annualized basis basis points regular basis weekly basis absolute basis

Phrases around this combination

Other phrases discovered around basis:

basis points 20×

Just before it in the phrase index:

basis points basic structure basic income basic education bases loaded based upon

Just after it in the phrase index:

basketball association basketball coach basketball league basketball player basketball team bass drum

Other phrases with the same pattern

Same start:

basis points

Corpus analysis

  • The phrase "basis vectors" has 2 words and usually appears in the middle in these examples. The average sentence has 28 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as combination of basis vectors or a, combination of basis vectors whose coefficients, vector, linear and orthogonal stand out.
  • In the phrase index, this combination connects with annualized basis, basis points, regular basis and basis points, linking the page to nearby combinations.

Example types with basis vectors

This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:

Short example sentences

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)

Longer example sentences

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 ).

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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.

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