Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
linear_combination_of_rows(v)
Return the linear combination of the rows of self given by the coefficients in the list v.
INPUT:
•v - a list of scalars. The length can be less than the number of rows of self but not greater.
OUTPUT:
The vector (or free module element) that is a linear combination of the rows of self. If the list of scalars
has fewer entries than the number of rows, additional zeros are appended to the list until it has as many
entries as the number of rows.
EXAMPLES:
sage: a = matrix(ZZ,2,3,range(6)); a
[0 1 2]
[3 4 5]
sage: a.linear_combination_of_rows([1,2])
(6, 9, 12)
sage: a.linear_combination_of_rows([0,0])
(0, 0, 0)
sage: a.linear_combination_of_rows([1/2,2/3])
(2, 19/6, 13/3)
The list v can be anything that is iterable. Perhaps most naturally, a vector may be used.
sage: v = vector(ZZ, [1,2])
sage: a.linear_combination_of_rows(v)
(6, 9, 12)
We check that a matrix with no rows behaves properly.
sage: matrix(QQ,0,2).linear_combination_of_rows([])
(0, 0)
The object returned is a vector, or a free module element.
sage: B = matrix(ZZ, 4, 3, range(12))
sage: w = B.linear_combination_of_rows([-1,2,-3,4])
sage: w
(24, 26, 28)
sage: w.parent()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
sage: x = B.linear_combination_of_rows([1/2,1/3,1/4,1/5])
sage: x
(43/10, 67/12, 103/15)
sage: x.parent()
Vector space of dimension 3 over Rational Field
The length of v can be less than the number of rows, but not greater.
sage: A = matrix(QQ,3,4,range(12))
sage: A.linear_combination_of_rows([2,3])
(12, 17, 22, 27)
sage: A.linear_combination_of_rows([1,2,3,4])
Traceback (most recent call last):
...
ValueError: length of v must be at most the number of rows of self
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