Accelerating the pace of engineering and science

# linalg::intBasis

Basis for the intersection of vector spaces

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```linalg::intBasis(S1, S2, …)
```

## Description

linalg::intBasis( S1, S2, ...) returns a basis for the intersection of the vector spaces spanned by the vectors in S1, S2, ….

The domain type of the vectors of the returned set is the domain type of the first parameter S1.

A basis for the zero-dimensional space is the empty set or empty 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 in 2:

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

A basis for the vector space V1V2V3 with

• V1 generated by

• V2 generated by

• V3 generated by

is:

`linalg::intBasis([v1, v2, v3], [v1, v3], [v1 + v2, v2, v1 + v3])`

### Example 2

The intersection of the two vector spaces spanned by the vectors in S1 and S2, respectively:

```S1 := {matrix([[1, 0, 1, 0]]), matrix([[0, 1, 0, 1]])};
S2 := {matrix([[1, 2, 1, 1]]), matrix([[-1, -2, 1, 0]])}```

is the zero-dimensional space:

`linalg::intBasis(S1, S2)`

## Parameters

 S1, S2, … Either sets or lists of n-dimensional vectors (a vector is an 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.