Getting started

>>> from ablina import *

Define a Vector Space

To define a subspace of R^n or C^n, use fn

>>> V = fn("V", R, 3)
>>> print(V.info())

V (Subspace of R^3)
-------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Dimension  3
Vector     [c0, c1, c2]

You can provide a list of constraints

>>> U = fn("U", R, 3, constraints=["v0 == 0", "2*v1 == v2"])
>>> print(U.info())

U (Subspace of R^3)
-------------------
Field      R
Identity   [0, 0, 0]
Basis      [[0, 1, 2]]
Dimension  1
Vector     [0, c0, 2*c0]

Or specify a basis

>>> W = fn("W", R, 3, basis=[[1, 0, 0], [0, 1, 0]])
>>> print(W.info())

W (Subspace of R^3)
-------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, 0], [0, 1, 0]]
Dimension  2
Vector     [c0, c1, 0]

Operations with Vectors

Check whether a vector is an element of a vector space

>>> [1, 2, 0] in W

True

>>> [1, 2, 1] in W

False

Generate a random vector from a vector space

>>> U.vector()

[0, 2, 4]

>>> U.vector(arbitrary=True)

[0, c0, 2*c0]

Find the coordinate vector representation of a vector

>>> W.to_coordinate([1, 2, 0])

[1, 2]

>>> W.from_coordinate([1, 2])

[1, 2, 0]

>>> W.to_coordinate([1, 2, 0], basis=[[1, 1, 0], [1, -1, 0]])

[3/2, -1/2]

Check whether a list of vectors is linearly independent

>>> V.is_independent([1, 1, 0], [1, 0, 0])

True

>>> V.is_independent([1, 2, 3], [2, 4, 6])

False

Operations on Vector Spaces

Check for equality of two vector spaces

>>> U == W

False

Check whether a vector space is a subspace of another

>>> V.is_subspace(U)

True

>>> U.is_subspace(V)

False

Take the sum of two vector spaces

>>> X = U.sum(W)
>>> print(X.info())

U + W (Subspace of R^3)
-----------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Dimension  3
Vector     [c0, c1, c2]

Take the intersection of two vector spaces

>>> X = U.intersection(W)
>>> print(X.info())

U ∩ W (Subspace of R^3)
-----------------------
Field      R
Identity   [0, 0, 0]
Basis      []
Dimension  0
Vector     [0, 0, 0]

Take the span of a list of vectors

>>> S = V.span("S", [1, 2, 3], [4, 5, 6])
>>> print(S.info())

S (Subspace of R^3)
-------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, -1], [0, 1, 2]]
Dimension  2
Vector     [c0, c1, -c0 + 2*c1]

Define a Linear Map

>>> def mapping(vec):
>>>     return [vec[0], vec[1], 0]
>>>
>>> T = LinearMap("T", domain=V, codomain=W, mapping=mapping)
>>> print(T.info())

T : V → W
---------
Field        R
Rank         2
Nullity      1
Injective?   False
Surjective?  True
Bijective?   False
Matrix       [[1, 0, 0], [0, 1, 0]]

>>> T([0, 0, 0])

[0, 0, 0]

>>> T([1, 2, 3])

[1, 2, 0]

Operations with Linear Maps

Find the image of a linear map

>>> im = T.image()
>>> print(im.info())

im(T) (Subspace of R^3)
-----------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, 0], [0, 1, 0]]
Dimension  2
Vector     [c0, c1, 0]

Find the kernel of a linear map

>>> ker = T.kernel()
>>> print(ker.info())

ker(T) (Subspace of R^3)
------------------------
Field      R
Identity   [0, 0, 0]
Basis      [[0, 0, 1]]
Dimension  1
Vector     [0, 0, c0]

Define an Inner Product

Here we define the standard dot product

>>> def mapping(vec1, vec2):
>>>     return sum(i * j for i, j in zip(vec1, vec2))
>>>
>>> dot = InnerProduct("dot", vectorspace=V, mapping=mapping)
>>> print(dot.info())

dot : V × V → R
---------------
Orthonormal Basis  [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Matrix             [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

>>> dot([1, 2, 3], [1, 2, 3])

14

Operations with Inner Products

Compute the norm of a vector

>>> dot.norm([1, 2, 3])

sqrt(14)

Check whether a list of vectors is pairwise orthogonal

>>> dot.is_orthogonal([1, 2, 3], [4, 5, 6])

False

>>> dot.is_orthogonal([0, 0, 0], [1, 2, 3])

True

Check whether a list of vectors is orthonormal

>>> dot.is_orthonormal([1, 0, 0], [0, 1, 0], [0, 0, 1])

True

Take the orthogonal complement of a vector space

>>> X = dot.ortho_complement(U)
>>> print(X.info())

perp(U) (Subspace of R^3)
-------------------------
Field      R
Identity   [0, 0, 0]
Basis      [[1, 0, 0], [0, 1, -1/2]]
Dimension  2
Vector     [c0, c1, -c1/2]

Define a Linear Operator

>>> def mapping(vec):
>>>     return [vec[0], 2*vec[1], 3*vec[2]]
>>>
>>> T = LinearOperator("T", vectorspace=V, mapping=mapping)
>>> print(T.info())

T : V → V
---------
Field        R
Rank         3
Nullity      0
Injective?   True
Surjective?  True
Bijective?   True
Matrix       [[1, 0, 0], [0, 2, 0], [0, 0, 3]]

>>> T([1, 1, 1])

[1, 2, 3]

Operations with Linear Operators

Given an inner product, check whether a linear operator

>>> T.is_symmetric(dot)

True

>>> T.is_hermitian(dot)

True

>>> T.is_orthogonal(dot)

False

>>> T.is_unitary(dot)

False

>>> T.is_normal(dot)

True