Vector

from abel.linalg.vector import Vector

a, b = Vector([1, 2]), Vector([3, 4])
c, d = Vector([1, 2, 3]), Vector([4, 5, 6])

assert a.shape == b.shape == (1, 2)
assert c.shape == d.shape == (1, 3)

Addition

assert a + a == Vector([2, 4])
assert a + b == Vector([4, 6])
assert b + b == Vector([6, 8])

Subtraction

assert a - b == Vector([-2, -2])
assert b - a == Vector([2, 2])

Scaling

assert a * 5 == Vector([5, 10])
assert 5 * a == Vector([5, 10])
assert a / 2 == Vector([0.5, 1.0])

Dot (inner) product

assert a @ b == 11
assert a @ a == 5

Norm (length)

assert a.norm() - 2.236 < 0.001
assert b.norm() - 5 < 0.001

Angle

assert Vector.angle(a, a) < 0.001
assert Vector.angle(a, b) - 0.1799 < 0.001
assert Vector.angle(a, b) == Vector.angle(b, a)

Vector projection

assert Vector.proj(a, a) == a
assert Vector.proj(a, b) == Vector([1.32, 1.76])
assert Vector.proj(b, a) == Vector([2.2, 4.4])

Scalar projection

assert Vector.scalproj(a, b) - 4.919 < 0.01
assert Vector.scalproj(b, a) - 2.2 < 0.01
assert Vector.scalproj(a, a) - 2.236 < 0.01
assert Vector.scalproj(b, b) - 5 < 0.1

Cross product

assert Vector.cross(c, d) == Vector([-3, 6, -3])

Average (arithmetic mean)

assert Vector.average(Vector([2, 1]), Vector([4, 2])) == Vector([3.0, 1.5])

Collinearity

Vectors are collinear iff one is a scalar multiple of the other.

assert Vector.collinear(Vector([2, 1]), Vector([4, 2]))
assert Vector.collinear(Vector([-3, 4, 1]), Vector([-15, 20, 5]))
assert not Vector.collinear(Vector([0, 1]), Vector([1, 0]))

Linear independence

A set of vectors is linearly independent iff all vectors in it are pairwise non-collinear.

assert Vector.linindep(Vector([0, 1]), Vector([1, 0]))
assert Vector.linindep(Vector([1, 1]), Vector([2, 1]))
assert not Vector.linindep(Vector([1, 2, 3]), Vector([0, 0, 1]), Vector([0, 0, 2]))