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Semantic Definition of a Vector Space
A vector space — also known as a linear space — is an abstract mathematical structure in which there exist objects called vectors. Two operations are defined for these objects: addition with each other and multiplication by numbers (scalars), with all operations obeying a strict set of rules — the eight axioms.
The most surprising thing is that this abstraction turned out to be incredibly universal: a wide range of phenomena in nature—forces, velocities, electromagnetic fields, as well as digital signals, images, and even texts—unexpectedly obey the same simple rules of linearity as vector spaces. Thanks to this, they form the basis of physics, computer graphics, data analysis, machine learning, and many other fields.
But it is important to remember: not all processes and objects in the world “add up” to a vector space. This model works only where all eight axioms are satisfied. Beyond these rules, there are other branches of mathematics and various laws.
Formal Definition of a Vector Space
A vector space (or linear space) over a field of scalars \(K\) (for example, the field of real numbers \(\mathbb{R}\) or complex numbers \(\mathbb{C}\)) is a set \(V\), whose elements are called vectors, with two operations:
- Vector addition: For any two vectors \(\mathbf{u}, \mathbf{v} \in V\), there is a defined operation of addition \(\mathbf{u} + \mathbf{v}\), whose result also belongs to \(V\).
- Scalar multiplication: For any vector \(\mathbf{v} \in V\) and scalar \(\alpha \in K\), there is a defined operation \(\alpha \mathbf{v}\), whose result also belongs to \(V\).
These operations must satisfy the following axioms:
Vector Space Axioms
- Associativity of addition: \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\) for any \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\).
- Commutativity of addition: \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\) for any \(\mathbf{u}, \mathbf{v} \in V\).
- Existence of a zero element: There exists a vector \(\mathbf{0} \in V\), called the zero vector, such that \(\mathbf{v} + \mathbf{0} = \mathbf{v}\) for any \(\mathbf{v} \in V\).
- Existence of additive inverses: For each \(\mathbf{v} \in V\), there exists a vector \(-\mathbf{v} \in V\) such that \(\mathbf{v} + (-\mathbf{v}) = \mathbf{0}\).
- Associativity of scalar multiplication: \(\alpha (\beta \mathbf{v}) = (\alpha \beta) \mathbf{v}\) for any \(\alpha, \beta \in K\) and \(\mathbf{v} \in V\).
- Distributivity with respect to vector addition: \(\alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v}\) for any \(\alpha \in K\) and \(\mathbf{u}, \mathbf{v} \in V\).
- Distributivity with respect to scalar addition: \((\alpha + \beta) \mathbf{v} = \alpha \mathbf{v} + \beta \mathbf{v}\) for any \(\alpha, \beta \in K\) and \(\mathbf{v} \in V\).
- Identity element of scalar multiplication: \(1 \cdot \mathbf{v} = \mathbf{v}\), where \(1\) is the unity of the field \(K\).
Why these eight axioms? This is not an arbitrary choice but a reflection of the structures found in physics and mathematics, where addition and scaling work “honestly” and symmetrically. But as soon as at least one of these axioms is violated, we leave the realm of vector spaces and enter other, more complex or less orderly systems. Therefore, the axioms of vector spaces are not a dogma, but a boundary of applicability of the model.
Examples of Vector Spaces
- Euclidean space \(\mathbb{R}^n\):
The set of \(n\)-dimensional vectors with coordinates from \(\mathbb{R}\), where addition and scalar multiplication are defined component-wise. For example, for \(\mathbb{R}^2\):- Vectors: \((x_1, y_1),\ (x_2, y_2)\).
- Addition: \((x_1, y_1) + (x_2, y_2) = (x_1 + x_2,\ y_1 + y_2)\).
- Scalar multiplication: \(\alpha (x, y) = (\alpha x,\ \alpha y)\).
- Space of polynomials \(P_n\):
The set of all polynomials of degree at most \(n\) with coefficients from \(K\). For example, \(P_2\) includes all polynomials of the form \(a + bx + cx^2\). - Space of matrices \(M_{m \times n}\):
The set of all matrices of size \(m \times n\) with elements from \(K\), where addition and scalar multiplication are performed element-wise. - Space of functions:
The set of all continuous functions on the interval \([a, b]\), where addition and scalar multiplication are defined as
\((f + g)(x) = f(x) + g(x)\) and \((\alpha f)(x) = \alpha f(x)\).
Basic Properties of Vector Spaces
- Basis: A set of vectors that is linearly independent and generates the entire space. For example, in \(\mathbb{R}^2\), a basis can be \({(1, 0),\ (0, 1)}\).
- Dimension: The number of vectors in a basis. For example, the dimension of \(\mathbb{R}^n\) is \(n\).
- Linear combination: A vector obtained as \(\alpha_1 \mathbf{v}_1 + \alpha_2 \mathbf{v}_2 + \dots + \alpha_n \mathbf{v}_n\), where \(\mathbf{v}_i \in V\), \(\alpha_i \in K\).
- Subspace: A subset of \(V\) that itself forms a vector space with the same operations.
Applications of Vector Spaces
Vector spaces are used in:
- Linear algebra: For solving systems of linear equations and matrix analysis.
- Physics: For describing physical quantities such as velocity, acceleration, and forces.
- Machine learning: Vectors represent data, and algorithms such as SVM or PCA rely on the properties of vector spaces.
- Computer graphics: For working with coordinates, transformations, and rendering.
Where Do the Axioms Not Work?
It is important to remember that not all systems in mathematics and nature are vector spaces. For example, in some physical or social systems, addition is not commutative, there is no neutral element, or there is no scalar multiplication operation. In such cases, one must go beyond linear algebra and develop new structures—groups, rings, topological spaces, and so on. This expands our mathematical view of the world and reminds us that universal models do not exist.
Conclusion
A vector space is not just a mathematical construct but a universal language for describing structures in which simple and powerful laws of linearity apply. Where the axioms hold, we have tools capable of uniting very different phenomena—from geometry to machine learning. But it is important to remember: the power of this model lies in its boundaries. As soon as one of the axioms is violated, a new, different world of mathematics arises, requiring other tools and approaches. That is why understanding vector spaces is the first step to mastering the immense diversity of mathematical worlds, where linearity is just one of many possible symphonies.