Vector spaces are fundamental algebraic structures in linear algebra. They consist of a set of vectors and a set of scalars (typically real or complex numbers) that satisfy a specific set of axioms governing vector addition and scalar multiplication.
Understanding vector spaces is crucial for grasping concepts like linear independence, basis, dimension, and linear transformations, which are foundational to many areas of mathematics, physics, engineering, computer science, and machine learning.
A vector space V over a field F (of scalars) is a set of objects called vectors, equipped with two operations:
These operations must satisfy the following ten axioms (for all vectors u, v, w in V and all scalars c, d in F):
Common examples of vector spaces include ℝn (n-dimensional real coordinate space), the set of all m x n matrices, the set of all real-valued functions, and the set of polynomials of degree at most n.
A subset W of a vector space V is a subspace of V if W itself is a vector space under the addition and scalar multiplication operations defined on V.
To verify if W is a subspace, one typically checks three conditions:
Examples include lines and planes passing through the origin in ℝ³.
A linear combination of vectors v1, v2, ..., vk in a vector space V is any vector of the form:
where c1, c2, ..., ck are scalars.
The span of a set of vectors S = {v1, ..., vk}, denoted span(S), is the set of all possible linear combinations of these vectors. The span of any set of vectors in V forms a subspace of V.
A set of vectors S = {v1, ..., vk} in a vector space V is linearly independent if the only solution to the equation:
is c1 = c2 = ... = ck = 0.
If there exist scalars, not all zero, for which the equation holds, then the set of vectors is linearly dependent. This means at least one vector in the set can be expressed as a linear combination of the others.
A basis for a vector space V is a set of vectors B = {b1, ..., bn} in V that satisfies two conditions:
Every vector in V can be expressed uniquely as a linear combination of basis vectors.
The dimension of a vector space V, denoted dim(V), is the number of vectors in any basis for V. If a vector space has a basis consisting of a finite number of vectors, it is called finite-dimensional; otherwise, it is infinite-dimensional.
For example, the standard basis for ℝn consists of n vectors, so dim(ℝn) = n.
If B = {b1, ..., bn} is an ordered basis for a vector space V, then any vector v in V can be written uniquely as:
The scalars c1, ..., cn are called the coordinates of v with respect to the basis B. The coordinate vector of v relative to B is [v]B = (c1, ..., cn)T.
For an m x n matrix A, there are four fundamental subspaces:
The Rank-Nullity Theorem states: rank(A) + dim(N(A)) = n (number of columns of A).
Implementing vector spaces involves creating a set of vectors and a set of scalars that satisfy the vector space axioms. This can be done in various programming languages such as Python, MATLAB, or Java.
In Python, for example, you can use lists to represent vectors and perform operations like vector addition and scalar multiplication. Here's a simple implementation:
class VectorSpace:
def __init__(self, vectors):
self.vectors = vectors
def add(self, v1, v2):
return [a + b for a, b in zip(v1, v2)]
def scalar_multiply(self, c, v):
return [c * a for a in v]
def is_vector_space(self):
# Implement the vector space axioms
pass
State the two main properties a set of vectors must satisfy to be a basis.
Analyze if this set meets vector space axioms.