The type of vectors in the column space.
The type of matrices in this matrix space.
A matrix in this matrix space.
The type of vectors in the row space.
The type of scalars in this matrix space.
The type of scalars in this matrix space.
The type of matrix transposes.
The type of vectors in this matrix space; equivalent to the type of matrices.
The type of vectors in this matrix space; equivalent to the type of matrices.
A vector in this vector space.
Returns the row space.
Returns the scalar set of this vector space.
Returns the scalar set of this vector space.
Returns the transpose of this matrix space.
Returns a new matrix with the given row-major entries.
Returns a new matrix with the given columns.
Returns the matrix product of the first matrix, whose column space equals this column space, times the second matrix, whose row space equals this row space, where the row space of the first matrix equals the column space of the second matrix.
Returns a new matrix with the given rows.
Returns the identity matrix of this matrix space, if one exists.
Returns the additive identity of this matrix space.
Returns the additive identity of this matrix space.
An abstract M by N matrix space over a ring. Matrix spaces describe linear maps between vector spaces relative to the vector spaces' assumed bases. Matrix addition associates and commutes, and scalar multiplication associates, commutes, and distributes over matrix addition and scalar addition. Matrix and vector multiplication also both associate and distribute over matrix addition. Vectors in the row space multiply as columns on the right, and vectors in the column space multiply as rows on the left. Addition and scalar multiplication both have an identity element, and every matrix has an additive inverse. To the extent practicable, the following axioms should hold.
Axioms for matrix addition:
this
, then their sum 𝐀 + 𝐁 is also a matrix inthis
.this
.this
.this
has a matrixzero
such thatzero
+ 𝐀 == 𝐀 for every matrix 𝐀 inthis
.this
corresponds a matrix -𝐀 inthis
such that 𝐀 + (-𝐀) ==zero
.Axioms for scalar multiplication:
this
and 𝐀 is a matrix inthis
, then their product 𝑥 *: 𝐀 is also a matrix inthis
.this
.Scalar
has an elementunit
such thatunit
*: 𝐀 == 𝐀 for every matrix 𝐀 inthis
.Axioms for vector multiplication:
this
and 𝐯 is a vector in the row space of 𝐀, then their product 𝐀 :⋅ 𝐯 is a vector in the column space of 𝐀.T
for every matrix 𝐀 and every vector 𝐯 in the row space of 𝐀.Axioms for matrix multiplication:
T
== 𝐁.T
⋅ 𝐀.T
for all matrices 𝐀, 𝐁 where the row space of 𝐀 equals the column space of 𝐁.Distributive laws:
this
.this
.0.1
0.0