Morphisms of vector spaces (linear transformations)#
AUTHOR:
Rob Beezer: (2011-06-29)
A vector space morphism is a homomorphism between vector spaces, better known as a linear transformation. These are a specialization of Sage’s free module homomorphisms. (A free module is like a vector space, but with scalars from a ring that may not be a field.) So references to free modules in the documentation or error messages should be understood as simply reflecting a more general situation.
Creation#
The constructor linear_transformation()
is designed to accept a
variety of inputs that can define a linear transformation. See the
documentation of the function for all the possibilities. Here we give two.
First a matrix representation. By default input matrices are understood to act on vectors placed to left of the matrix. Optionally, an input matrix can be described as acting on vectors placed to the right.
sage: A = matrix(QQ, [[-1, 2, 3], [4, 2, 0]])
sage: phi = linear_transformation(A)
sage: phi
Vector space morphism represented by the matrix:
[-1 2 3]
[ 4 2 0]
Domain: Vector space of dimension 2 over Rational Field
Codomain: Vector space of dimension 3 over Rational Field
sage: phi([2, -3])
(-14, -2, 6)
A symbolic function can be used to specify the “rule” for a linear transformation, along with explicit descriptions of the domain and codomain.
sage: # needs sage.symbolic
sage: F = Integers(13)
sage: D = F^3
sage: C = F^2
sage: x, y, z = var('x y z')
sage: f(x, y, z) = [2*x + 3*y + 5*z, x + z]
sage: rho = linear_transformation(D, C, f)
sage: f(1, 2, 3)
(23, 4)
sage: rho([1, 2, 3])
(10, 4)
A “vector space homspace” is the set of all linear transformations
between two vector spaces. Various input can be coerced into a
homspace to create a linear transformation. See
sage.modules.vector_space_homspace
for more.
sage: D = QQ^4
sage: C = QQ^2
sage: hom_space = Hom(D, C)
sage: images = [[1, 3], [2, -1], [4, 0], [3, 7]]
sage: zeta = hom_space(images)
sage: zeta
Vector space morphism represented by the matrix:
[ 1 3]
[ 2 -1]
[ 4 0]
[ 3 7]
Domain: Vector space of dimension 4 over Rational Field
Codomain: Vector space of dimension 2 over Rational Field
A homomorphism may also be created via a method on the domain.
sage: # needs sage.rings.number_field sage.symbolic
sage: F = QQ[sqrt(3)]
sage: a = F.gen(0)
sage: D = F^2
sage: C = F^2
sage: A = matrix(F, [[a, 1], [2*a, 2]])
sage: psi = D.hom(A, C)
sage: psi
Vector space morphism represented by the matrix:
[ sqrt3 1]
[2*sqrt3 2]
Domain: Vector space of dimension 2 over Number Field in sqrt3
with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?
Codomain: Vector space of dimension 2 over Number Field in sqrt3
with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?
sage: psi([1, 4])
(9*sqrt3, 9)
Properties#
Many natural properties of a linear transformation can be computed.
Some of these are more general methods of objects in the classes
sage.modules.free_module_morphism.FreeModuleMorphism
and
sage.modules.matrix_morphism.MatrixMorphism
.
Values are computed in a natural way, an inverse image of an
element can be computed with the lift()
method, when the inverse
image actually exists.
sage: A = matrix(QQ, [[1,2], [2,4], [3,6]])
sage: phi = linear_transformation(A)
sage: phi([1,2,0])
(5, 10)
sage: phi.lift([10, 20])
(10, 0, 0)
sage: phi.lift([100, 100])
Traceback (most recent call last):
...
ValueError: element is not in the image
Images and pre-images can be computed as vector spaces.
sage: A = matrix(QQ, [[1,2], [2,4], [3,6]])
sage: phi = linear_transformation(A)
sage: phi.image()
Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[1 2]
sage: phi.inverse_image( (QQ^2).span([[1,2]]) )
Vector space of degree 3 and dimension 3 over Rational Field
Basis matrix:
[1 0 0]
[0 1 0]
[0 0 1]
sage: phi.inverse_image( (QQ^2).span([[1,1]]) )
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1/3]
[ 0 1 -2/3]
Injectivity and surjectivity can be checked.
sage: A = matrix(QQ, [[1,2], [2,4], [3,6]])
sage: phi = linear_transformation(A)
sage: phi.is_injective()
False
sage: phi.is_surjective()
False
Restrictions and representations#
It is possible to restrict the domain and codomain of a linear transformation to make a new linear transformation. We will use those commands to replace the domain and codomain by equal vector spaces, but with alternate bases. The point here is that the matrix representation used to represent linear transformations are relative to the bases of both the domain and codomain.
sage: A = graphs.PetersenGraph().adjacency_matrix() # needs sage.graphs
sage: V = QQ^10
sage: phi = linear_transformation(V, V, A) # needs sage.graphs
sage: phi # needs sage.graphs
Vector space morphism represented by the matrix:
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Domain: Vector space of dimension 10 over Rational Field
Codomain: Vector space of dimension 10 over Rational Field
sage: # needs sage.graphs
sage: B1 = [V.gen(i) + V.gen(i+1) for i in range(9)] + [V.gen(9)]
sage: B2 = [V.gen(0)] + [-V.gen(i-1) + V.gen(i) for i in range(1,10)]
sage: D = V.subspace_with_basis(B1)
sage: C = V.subspace_with_basis(B2)
sage: rho = phi.restrict_codomain(C)
sage: zeta = rho.restrict_domain(D)
sage: zeta
Vector space morphism represented by the matrix:
[6 5 4 3 3 2 1 0 0 0]
[6 5 4 3 2 2 2 1 0 0]
[6 6 5 4 3 2 2 2 1 0]
[6 5 5 4 3 2 2 2 2 1]
[6 4 4 4 3 3 3 3 2 1]
[6 5 4 4 4 4 4 4 3 1]
[6 6 5 4 4 4 3 3 3 2]
[6 6 6 5 4 4 2 1 1 1]
[6 6 6 6 5 4 3 1 0 0]
[3 3 3 3 3 2 2 1 0 0]
Domain: Vector space of degree 10 and dimension 10 over Rational Field
User basis matrix:
[1 1 0 0 0 0 0 0 0 0]
[0 1 1 0 0 0 0 0 0 0]
[0 0 1 1 0 0 0 0 0 0]
[0 0 0 1 1 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1]
Codomain: Vector space of degree 10 and dimension 10 over Rational Field
User basis matrix:
[ 1 0 0 0 0 0 0 0 0 0]
[-1 1 0 0 0 0 0 0 0 0]
[ 0 -1 1 0 0 0 0 0 0 0]
[ 0 0 -1 1 0 0 0 0 0 0]
[ 0 0 0 -1 1 0 0 0 0 0]
[ 0 0 0 0 -1 1 0 0 0 0]
[ 0 0 0 0 0 -1 1 0 0 0]
[ 0 0 0 0 0 0 -1 1 0 0]
[ 0 0 0 0 0 0 0 -1 1 0]
[ 0 0 0 0 0 0 0 0 -1 1]
An endomorphism is a linear transformation with an equal domain and codomain, and here each needs to have the same basis. We are using a matrix that has well-behaved eigenvalues, as part of showing that these do not change as the representation changes.
sage: # needs sage.graphs
sage: A = graphs.PetersenGraph().adjacency_matrix()
sage: V = QQ^10
sage: phi = linear_transformation(V, V, A)
sage: phi.eigenvalues() # needs sage.rings.number_field
[3, -2, -2, -2, -2, 1, 1, 1, 1, 1]
sage: B1 = [V.gen(i) + V.gen(i+1) for i in range(9)] + [V.gen(9)]
sage: C = V.subspace_with_basis(B1)
sage: zeta = phi.restrict(C)
sage: zeta
Vector space morphism represented by the matrix:
[ 1 0 1 -1 2 -1 2 -2 2 -2]
[ 1 0 1 0 0 0 1 0 0 0]
[ 0 1 0 1 0 0 0 1 0 0]
[ 1 -1 2 -1 2 -2 2 -2 3 -2]
[ 2 -2 2 -1 1 -1 1 0 1 0]
[ 1 0 0 0 0 0 0 1 1 0]
[ 0 1 0 0 0 1 -1 1 0 2]
[ 0 0 1 0 0 2 -1 1 -1 2]
[ 0 0 0 1 0 1 1 0 0 0]
[ 0 0 0 0 1 -1 2 -1 1 -1]
Domain: Vector space of degree 10 and dimension 10 over Rational Field
User basis matrix:
[1 1 0 0 0 0 0 0 0 0]
[0 1 1 0 0 0 0 0 0 0]
[0 0 1 1 0 0 0 0 0 0]
[0 0 0 1 1 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1]
Codomain: Vector space of degree 10 and dimension 10 over Rational Field
User basis matrix:
[1 1 0 0 0 0 0 0 0 0]
[0 1 1 0 0 0 0 0 0 0]
[0 0 1 1 0 0 0 0 0 0]
[0 0 0 1 1 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 1 0 0 0]
[0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1]
sage: zeta.eigenvalues() # needs sage.rings.number_field
[3, -2, -2, -2, -2, 1, 1, 1, 1, 1]
Equality#
Equality of linear transformations is a bit nuanced. The equality operator
==
tests if two linear transformations have equal matrix representations,
while we determine if two linear transformations are the same function with the
.is_equal_function()
method. Notice in this example that the function
never changes, just the representations.
sage: f = lambda x: vector(QQ, [x[1], x[0]+x[1], x[0]])
sage: H = Hom(QQ^2, QQ^3)
sage: phi = H(f)
sage: rho = linear_transformation(QQ^2, QQ^3, matrix(QQ,2, 3, [[0,1,1], [1,1,0]]))
sage: phi == rho
True
sage: U = (QQ^2).subspace_with_basis([[1, 2], [-3, 1]])
sage: V = (QQ^3).subspace_with_basis([[0, 1, 0], [2, 3, 1], [-1, 1, 6]])
sage: K = Hom(U, V)
sage: zeta = K(f)
sage: zeta == phi
False
sage: zeta.is_equal_function(phi)
True
sage: zeta.is_equal_function(rho)
True
- class sage.modules.vector_space_morphism.VectorSpaceMorphism(homspace, A, side='left')#
Bases:
FreeModuleMorphism
Create a linear transformation, a morphism between vector spaces.
INPUT:
homspace
- a homspace (of vector spaces) to serve as a parent for the linear transformation and a home for the domain and codomain of the morphismA
- a matrix representing the linear transformation, which will act on vectors placed to the left of the matrix
EXAMPLES:
Nominally, we require a homspace to hold the domain and codomain and a matrix representation of the morphism (linear transformation).
sage: from sage.modules.vector_space_homspace import VectorSpaceHomspace sage: from sage.modules.vector_space_morphism import VectorSpaceMorphism sage: H = VectorSpaceHomspace(QQ^3, QQ^2) sage: A = matrix(QQ, 3, 2, range(6)) sage: zeta = VectorSpaceMorphism(H, A) sage: zeta Vector space morphism represented by the matrix: [0 1] [2 3] [4 5] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space of dimension 2 over Rational Field
See the constructor,
sage.modules.vector_space_morphism.linear_transformation()
for another way to create linear transformations.The
.hom()
method of a vector space will create a vector space morphism.sage: V = QQ^3; W = V.subspace_with_basis([[1,2,3], [-1,2,5/3], [0,1,-1]]) sage: phi = V.hom(matrix(QQ, 3, range(9)), codomain=W) # indirect doctest sage: type(phi) <class 'sage.modules.vector_space_morphism.VectorSpaceMorphism'>
A matrix may be coerced into a vector space homspace to create a vector space morphism.
sage: from sage.modules.vector_space_homspace import VectorSpaceHomspace sage: H = VectorSpaceHomspace(QQ^3, QQ^2) sage: A = matrix(QQ, 3, 2, range(6)) sage: rho = H(A) # indirect doctest sage: type(rho) <class 'sage.modules.vector_space_morphism.VectorSpaceMorphism'>
- is_invertible()#
Determines if the vector space morphism has an inverse.
OUTPUT:
True
if the vector space morphism is invertible, otherwiseFalse
.EXAMPLES:
If the dimension of the domain does not match the dimension of the codomain, then the morphism cannot be invertible.
sage: V = QQ^3 sage: U = V.subspace_with_basis([V.0 + V.1, 2*V.1 + 3*V.2]) sage: phi = V.hom([U.0, U.0 + U.1, U.0 - U.1], U) sage: phi.is_invertible() False
An invertible linear transformation.
sage: A = matrix(QQ, 3, [[-3, 5, -5], [4, -7, 7], [6, -8, 10]]) sage: A.determinant() 2 sage: H = Hom(QQ^3, QQ^3) sage: rho = H(A) sage: rho.is_invertible() True
A non-invertible linear transformation, an endomorphism of a vector space over a finite field.
sage: # needs sage.rings.finite_rings sage: F.<a> = GF(11^2) sage: A = matrix(F, [[6*a + 3, 8*a + 2, 10*a + 3], ....: [2*a + 7, 4*a + 3, 2*a + 3], ....: [9*a + 2, 10*a + 10, 3*a + 3]]) sage: A.nullity() 1 sage: E = End(F^3) sage: zeta = E(A) sage: zeta.is_invertible() False
- sage.modules.vector_space_morphism.is_VectorSpaceMorphism(x)#
Returns
True
ifx
is a vector space morphism (a linear transformation).INPUT:
x
- anythingOUTPUT:
True
only ifx
is an instance of a vector space morphism, which are also known as linear transformations.EXAMPLES:
sage: V = QQ^2; f = V.hom([V.1,-2*V.0]) sage: sage.modules.vector_space_morphism.is_VectorSpaceMorphism(f) True sage: sage.modules.vector_space_morphism.is_VectorSpaceMorphism('junk') False
- sage.modules.vector_space_morphism.linear_transformation(arg0, arg1=None, arg2=None, side='left')#
Create a linear transformation from a variety of possible inputs.
FORMATS:
In the following,
D
andC
are vector spaces over the same field that are the domain and codomain (respectively) of the linear transformation.side
is a keyword that is either ‘left’ or ‘right’. When a matrix is used to specify a linear transformation, as in the first two call formats below, you may specify if the function is given by matrix multiplication with the vector on the left, or the vector on the right. The default is ‘left’. The matrix representation may be obtained as either version, no matter how it is created.linear_transformation(A, side='left')
Where
A
is a matrix. The domain and codomain are inferred from the dimension of the matrix and the base ring of the matrix. The base ring must be a field, or have its fraction field implemented in Sage.linear_transformation(D, C, A, side='left')
A
is a matrix that behaves as above. However, now the domain and codomain are given explicitly. The matrix is checked for compatibility with the domain and codomain. Additionally, the domain and codomain may be supplied with alternate (“user”) bases and the matrix is interpreted as being a representation relative to those bases.linear_transformation(D, C, f)
f
is any function that can be applied to the basis elements of the domain and that produces elements of the codomain. The linear transformation returned is the unique linear transformation that extends this mapping on the basis elements.f
may come from a function defined by a Pythondef
statement, or may be defined as alambda
function.Alternatively,
f
may be specified by a callable symbolic function, see the examples below for a demonstration.linear_transformation(D, C, images)
images
is a list, or tuple, of codomain elements, equal in number to the size of the basis of the domain. Each basis element of the domain is mapped to the corresponding element of theimages
list, and the linear transformation returned is the unique linear transformation that extends this mapping.
OUTPUT:
A linear transformation described by the input. This is a “vector space morphism”, an object of the class
sage.modules.vector_space_morphism
.EXAMPLES:
We can define a linear transformation with just a matrix, understood to act on a vector placed on one side or the other. The field for the vector spaces used as domain and codomain is obtained from the base ring of the matrix, possibly promoting to a fraction field.
sage: A = matrix(ZZ, [[1, -1, 4], [2, 0, 5]]) sage: phi = linear_transformation(A) sage: phi Vector space morphism represented by the matrix: [ 1 -1 4] [ 2 0 5] Domain: Vector space of dimension 2 over Rational Field Codomain: Vector space of dimension 3 over Rational Field sage: phi([1/2, 5]) (21/2, -1/2, 27) sage: B = matrix(Integers(7), [[1, 2, 1], [3, 5, 6]]) sage: rho = linear_transformation(B, side='right') sage: rho Vector space morphism represented by the matrix: [1 3] [2 5] [1 6] Domain: Vector space of dimension 3 over Ring of integers modulo 7 Codomain: Vector space of dimension 2 over Ring of integers modulo 7 sage: rho([2, 4, 6]) (2, 6)
We can define a linear transformation with a matrix, while explicitly giving the domain and codomain. Matrix entries will be coerced into the common field of scalars for the vector spaces.
sage: D = QQ^3 sage: C = QQ^2 sage: A = matrix([[1, 7], [2, -1], [0, 5]]) sage: A.parent() Full MatrixSpace of 3 by 2 dense matrices over Integer Ring sage: zeta = linear_transformation(D, C, A) sage: zeta.matrix().parent() Full MatrixSpace of 3 by 2 dense matrices over Rational Field sage: zeta Vector space morphism represented by the matrix: [ 1 7] [ 2 -1] [ 0 5] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space of dimension 2 over Rational Field
Matrix representations are relative to the bases for the domain and codomain.
sage: u = vector(QQ, [1, -1]) sage: v = vector(QQ, [2, 3]) sage: D = (QQ^2).subspace_with_basis([u, v]) sage: x = vector(QQ, [2, 1]) sage: y = vector(QQ, [-1, 4]) sage: C = (QQ^2).subspace_with_basis([x, y]) sage: A = matrix(QQ, [[2, 5], [3, 7]]) sage: psi = linear_transformation(D, C, A) sage: psi Vector space morphism represented by the matrix: [2 5] [3 7] Domain: Vector space of degree 2 and dimension 2 over Rational Field User basis matrix: [ 1 -1] [ 2 3] Codomain: Vector space of degree 2 and dimension 2 over Rational Field User basis matrix: [ 2 1] [-1 4] sage: psi(u) == 2*x + 5*y True sage: psi(v) == 3*x + 7*y True
Functions that act on the domain may be used to compute images of the domain’s basis elements, and this mapping can be extended to a unique linear transformation. The function may be a Python function (via
def
orlambda
) or a Sage symbolic function.sage: def g(x): ....: return vector(QQ, [2*x[0]+x[2], 5*x[1]]) sage: phi = linear_transformation(QQ^3, QQ^2, g) sage: phi Vector space morphism represented by the matrix: [2 0] [0 5] [1 0] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space of dimension 2 over Rational Field sage: f = lambda x: vector(QQ, [2*x[0]+x[2], 5*x[1]]) sage: rho = linear_transformation(QQ^3, QQ^2, f) sage: rho Vector space morphism represented by the matrix: [2 0] [0 5] [1 0] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space of dimension 2 over Rational Field sage: # needs sage.symbolic sage: x, y, z = var('x y z') sage: h(x, y, z) = [2*x + z, 5*y] sage: zeta = linear_transformation(QQ^3, QQ^2, h) sage: zeta Vector space morphism represented by the matrix: [2 0] [0 5] [1 0] Domain: Vector space of dimension 3 over Rational Field Codomain: Vector space of dimension 2 over Rational Field sage: phi == rho True sage: rho == zeta # needs sage.symbolic True
We create a linear transformation relative to non-standard bases, and capture its representation relative to standard bases. With this, we can build functions that create the same linear transformation relative to the nonstandard bases.
sage: u = vector(QQ, [1, -1]) sage: v = vector(QQ, [2, 3]) sage: D = (QQ^2).subspace_with_basis([u, v]) sage: x = vector(QQ, [2, 1]) sage: y = vector(QQ, [-1, 4]) sage: C = (QQ^2).subspace_with_basis([x, y]) sage: A = matrix(QQ, [[2, 5], [3, 7]]) sage: psi = linear_transformation(D, C, A) sage: rho = psi.restrict_codomain(QQ^2).restrict_domain(QQ^2) sage: rho.matrix() [ -4/5 97/5] [ 1/5 -13/5] sage: f = lambda x: vector(QQ, [(-4/5)*x[0] + (1/5)*x[1], (97/5)*x[0] + (-13/5)*x[1]]) sage: psi = linear_transformation(D, C, f) sage: psi.matrix() [2 5] [3 7] sage: # needs sage.symbolic sage: s, t = var('s t') sage: h(s, t) = [(-4/5)*s + (1/5)*t, (97/5)*s + (-13/5)*t] sage: zeta = linear_transformation(D, C, h) sage: zeta.matrix() [2 5] [3 7]
Finally, we can give an explicit list of images for the basis elements of the domain.
sage: # needs sage.rings.number_field sage: x = polygen(QQ) sage: F.<a> = NumberField(x^3 + x + 1) sage: u = vector(F, [1, a, a^2]) sage: v = vector(F, [a, a^2, 2]) sage: w = u + v sage: D = F^3 sage: C = F^3 sage: rho = linear_transformation(D, C, [u, v, w]) sage: rho.matrix() [ 1 a a^2] [ a a^2 2] [ a + 1 a^2 + a a^2 + 2] sage: C = (F^3).subspace_with_basis([u, v]) sage: D = (F^3).subspace_with_basis([u, v]) sage: psi = linear_transformation(C, D, [u+v, u-v]) sage: psi.matrix() [ 1 1] [ 1 -1]