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linalg::sumBasis

Basis for the sum of vector spaces

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Syntax

linalg::sumBasis(S1, S2, …)

Description

linalg::sumBasis( S1, S2, ...) returns a basis of the vector space V1 + V2 + …, where Vi denotes the vector space spanned by the vectors in Si.

To obtain an ordered basis, S1, S2, ... should be given as lists of vectors.

A basis of the zero-dimensional space is the empty set or list, respectively.

The given vectors must be defined over the same component ring, which must be a field, i.e., a domain of category Cat::Field.

Examples

Example 1

We define three vectors over :

MatQ := Dom::Matrix(Dom::Rational):
v1 := MatQ([[3, -2]]); v2 := MatQ([[1, 0]]); v3 := MatQ([[5, -3]])

A basis of the vector space V1 + V2 + V3 with

  • V1 generated by

  • V2 generated by

  • V3 generated by

is:

linalg::sumBasis([v1, v2, v3], [v1, v3], [v1 + v2, v2, v1 + v3])

Example 2

The following set of two vectors:

MatQ := Dom::Matrix(Dom::Rational):
S1 := {MatQ([1, 2, 3]), MatQ([-1, 0, 2])}

is a basis of a two-dimensional subspace of 3:

linalg::rank(S1)

The same holds for the following set:

S2 := {MatQ([0, 2, 3]), MatQ([2, 4, 6])};
linalg::rank(S2)

The sum of the corresponding two subspaces is the vector space 3:

Q3 := linalg::sumBasis(S1, S2)

Parameters

S1, S2, …

A set or list of vectors of the same dimension (a vector is a n×1 or 1 ×n matrix of a domain of category Cat::Matrix)

Return Values

Set or a list of vectors, according to the domain type of the parameter S1.

See Also

MuPAD Functions

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