Representation and Reconstruction

This page is a sub-page of our page on Vector spaces.

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Related KMR-pages:

Representation and Reconstruction of Linear Transformations

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Representation and Reconstruction of a Presentant with respect to a Background

Representation: \, [ \, p_{resentant} \, ]_{B_{ackground}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{ackground}}

Reconstruction: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{ackground}} \, \right)_{B_{ackground}} \mapsto \,\, p_{resentant}

/////// In Swedish:

Representation och Rekonstruktion av en Presentant med avseende på en Bakgrund

Representation: \, [ \, p_{resentant} \, ]_{B_{akgrund}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}}

Rekonstruktion: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}} \, \right)_{B_{akgrund}} \mapsto \,\, p_{resentant}

/////// Back to English:

Representation and Reconstruction of Vectors

The Representation and Reconstruction formulas for Numbers carry over verbatim to the corresponding formulas for Vectors, except for the replacement of the term \, B_{ase} \, by the term \, B_{asis} \, and the replacement of the term \, d_{igits} \, by the term \, c_{oordinates} \,.

Hence, just as a \, n_{umber} \, is represented by its \, d_{igits} \, with respect to a certain \, B_{ase} \, ,
a \, v_{ector} \, is represented by its \, c_{oordinates} \, with respect to a a certain \, B_{asis} \,.

Representation: [ \, v_{ector} \, ]_{B_{asis}} \mapsto \left<c_{oordinates}\right>_{B_{asis}}

Reconstruction: \left( \, \left< \, c_{oordinates} \, \right>_{B_{asis}} \, \right)_{B_{asis}} = v_{ector} \,

B_{asis} = \{ B_1, B_2, B_3, \cdots \}

v_{ector} \equiv {c_{oordinate}}_1 \, B_1 + {c_{oordinate}}_2 \, B_2 + {c_{oordinate}}_3 \, B_3 + \cdots \equiv

\,\,\,\,\,\,\,\,\,\,\,\,\,\, \equiv c_1 \, B_1 + c_2 \, B_2 + c_3 \, B_3 + \cdots \, .

Hence, the coordinates or coordinate vector of a vector with respect to a certain basis is the sequence of coefficients that is obtained by expressing the vector as a totally ordered linear combination of the linearly independent vectors in the basis.

IMPORTANT: Expressing the coordinates of a vector as a sequence requires that the corresponding basis-vectors have been ordered in some way. This ordering can be chosen arbitrarily, but it has to reappear in the ordering of the corresponding coordinates.

NOTATION: In our terminology, there is a difference between the terms \, B_{ase} \, and \, B_{asis} \, . A \, B_{ase} \, consists of one element, such as, for example, the chosen “base-number” for the digits-representation of a “number” in the position system, or the chosen musical “base-note” that specifies the \, K_{ey} \, in which a piece of tonal music is to be played.

In contrast, a \, B_{asis} \, normally consists of several elements. For example, in linear algebra (a.k.a. vector algebra), a basis for a vector space consists of a set of linearly independent vectors that span the entire space. The number of basis vectors for a vector space is called its dimension. The dimension of a vector space is well defined, since all bases for the same vector space have the same cardinality (= number of elements).

At the page Shift of Basis for a Vector Space it is demonstrated how the basis-shift formula for vectors can be expressed in terms of matrix algebra.

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