Key Properties and Real-Life Examples of Vector Spaces
What is Vector and Vector Space?
The vectors can be added as well as multiplied by scalars while preserving the ordinary arithmetic properties.
So, How do you Define a Vector Space?
A vector space is one in which the elements are sets of numbers themselves. Every element in a vector space is a list of objects with specific length, which we call vectors. The elements of a vector space are often referred to as n-tuples, where n is the specific length of each of the elements in the set. Matrix is another way of representing each element of a vector space of length n.
History
Historically, the first ideas relating to vector spaces came from analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Analytic geometry was founded by French Mathematicians René Descartes and Pierre de Fermat around 1636. They identified a solution to an equation of two variables with points on a plane curve. Bolzano introduced certain operations on points, lines and planes, which are called predecessors of vectors in order to achieve geometric solutions without using coordinates. The modern and more abstract treatment was formulated by Giuseppe Peano in 1888.
Vector spaces branches out from affine geometry, through the introduction of coordinates in the plane or three-dimensional space. In other words, vector spaces are mathematical objects. They abstractly capture the geometry and algebra of linear equations and are the central objects of study in linear algebra. They often appear throughout mathematics and physics.
Vector Addition
Vector addition is a way of combining two vectors, say u and v, into a single vector like this: u+v.
There are few conditions that must be satisfied by the operation of vector addition. They are:
Closure: If u and v are the vectors in V, the sum of u and v ( u+v) will belong to V.
Commutative Law: For all the vectors (u and v) in V, u + v is equal to v + u.
Associative Law: Vectors u, v, w in V, u + (v + w) is equal to (u + v) + w.
Additive Identity: The set V has an additive identity element which is usually denoted by 0, such that for any vector (v) in V, 0 + v = v and v + 0 = v.
Additive Inverses: For every vector v in V, the equations v + x = 0 and x + v = 0 contains a solution x in V which is called an additive inverse of v, and is denoted by - v.
Scalar Multiplication
Scalar multiplication is a way of combining a scalar k, along with a vector v, to end up with the vector kv. The operation of scalar multiplication can be explained between real numbers and vectors that must satisfy few conditions, they are:
1) Closure: If v is a vector in V and c in real numbers, the product c-v will belong to V.
2) Distributive Law: For the real number c and vectors u & v in V, c · (u + v) = c · u + c · v.
3) Distributive Law: For the real numbers c & d and vectors v in V, (c+d) · v = c · v + d · v
4) Associative Law: For the real numbers c & d and vectors v in V, c · (d · v) = (cd) · v
5) Unitary Law: For the vector v in V, 1 · v = v
Axioms for Vector Spaces
Vector space can be defined by ten axioms. Let x, y, & z be the elements of the vector space V and a & b be the elements of the field F.
Closed Under Addition: For every element x and y in V, x + y is also in V.
Closed Under Scalar Multiplication: For every element x in V and scalar a in F, ax is in V.
Commutativity of Addition: For every element x and y in V, x + y = y + x.
Associativity of Addition: For every element x, y, and z in V, (x + y) + z = x + (y + z).
Existence of the Additive Identity: There exists an element in V which is denoted as 0 such that x + 0 = x, for all x in V.
Existence of the Additive Inverse: For every element x in V, there exists another element in V that we can call -x such that x + (-x) = 0.
Existence of the Multiplicative Identity: There exists an element in F notated as 1 so that for all x in V, 1x = x.
Associativity of Scalar Multiplication: For every element x in V, and for each pair of elements a and b in F, (ab)x = a(bx).
Distribution of Elements to Scalars: For every element a in F and every pair of elements x and y in V, a(x + y) = ax + ay.
Distribution of Scalars to Elements: For every element x in V, and every pair of elements a and b in F, (a + b)x = ax + bx.
Vector Space Examples
Here are the spaces of n-tuples where each part of every element is a real number. The set of scalars are also the set of real numbers. Let's take a look at some key definitions.
Addition is explained as adding the corresponding parts of each element: (a, b, . . . ) + (c, d, .. . ) is equal to (a + c, b + d, .. .).
Scalar multiplication is explained as multiplying every part of the element by the scalar: a(b, c, . . . ) = (ab, ac, . . . ).
For these vector spaces the additive identity is the element (0, 0, 0, . . . , 0), where there are n 0s in this element.
For these vector spaces the multiplicative identity is the scalar 1 from the field of real numbers R.
The following are the basic vector space examples, but there is no proof that the space R3 is a vector space.
FAQs on Vector Space: Definition, Properties & Uses
1. What is a vector space in simple terms?
In simple terms, a vector space is a collection of objects called vectors, which can be added together and multiplied (or 'scaled') by numbers, called scalars. For a collection to be a true vector space, these operations of addition and scalar multiplication must follow a set of ten specific rules, or axioms. A common example that students learn is the set of all vectors in a 2D plane (denoted as ℝ²), where you can add vectors head-to-tail and stretch or shrink them by multiplying by a number.
2. What are the fundamental properties (axioms) that define a vector space?
A set V is called a vector space over a field F of scalars if it satisfies the following ten axioms for all vectors u, v, w in V and all scalars c, d in F:
- Closure under Addition: If u and v are in V, then u + v is also in V.
- Commutativity of Addition: u + v = v + u.
- Associativity of Addition: (u + v) + w = u + (v + w).
- Existence of an Additive Identity: There is a zero vector 0 in V such that v + 0 = v.
- Existence of an Additive Inverse: For every v in V, there is an inverse vector -v such that v + (-v) = 0.
- Closure under Scalar Multiplication: If c is a scalar and v is in V, then cv is also in V.
- Distributivity of Scalar over Vector Addition: c(u + v) = cu + cv.
- Distributivity of Scalar Addition over a Vector: (c + d)v = cv + dv.
- Associativity of Scalar Multiplication: c(dv) = (cd)v.
- Existence of a Scalar Identity: There is a scalar 1 such that 1v = v.
3. What are some common examples of a vector space that students encounter?
Students often work with vector spaces without knowing the formal term. Common examples include:
- Geometric Vectors: The set of all vectors in a 2D plane (ℝ²) or in 3D space (ℝ³) that you study in physics and maths are prime examples.
- The Set of Real Numbers: The set of all real numbers (ℝ) can be considered a vector space over itself.
- Polynomials: The set of all polynomials with real coefficients of a degree less than or equal to a specific number 'n' (denoted Pn) forms a vector space.
- Matrices: The set of all m x n matrices with real entries forms a vector space under standard matrix addition and scalar multiplication.
4. Are vectors in a vector space always arrows with a specific direction and magnitude?
No, this is a common misconception. While geometric arrows in 2D or 3D space are a very important and intuitive example of vectors, the concept of a 'vector' in a vector space is far more abstract. An object is a vector if it belongs to a set that satisfies the ten vector space axioms. This means a vector can be a polynomial, a matrix, a function, or even an infinite sequence of numbers. The idea of 'direction and magnitude' applies well to geometric vectors but not to these other abstract types.
5. How can you determine if a given set with defined operations forms a vector space?
To determine if a set V, with operations of addition and scalar multiplication, is a vector space, you must systematically check if all ten vector space axioms are satisfied. The process involves:
- Confirming that the set is closed under both vector addition and scalar multiplication.
- Verifying the associative, commutative, and distributive properties.
- Identifying a unique zero vector (the additive identity) within the set.
- Ensuring that for every vector in the set, a corresponding additive inverse also exists within the set.
If even one of these axioms fails, the set is not a vector space.
6. Why is the concept of a vector space so important in subjects like physics and engineering?
The concept of a vector space is crucial because it provides a single, unified framework to understand and manipulate many different systems that behave in a linear fashion. Its importance lies in its generality and applicability:
- In Physics: It is used to describe quantities like force, velocity, acceleration, and electric and magnetic fields. Quantum mechanics is fundamentally built on the principles of vector spaces.
- In Engineering: It is essential for solving systems of linear equations, signal processing, control theory, and structural analysis.
- In Computer Science: It is the foundation for computer graphics (representing 3D objects and transformations), machine learning (representing data points as vectors), and data compression.
By abstracting properties, we can apply the same powerful tools of linear algebra to solve problems in these diverse fields.
7. What is the key difference between a vector space and a ring in abstract algebra?
The key difference lies in the structures and operations involved. A vector space is defined over two sets: a set of vectors (V) and a field of scalars (F), with two operations connecting them: vector addition (V x V → V) and scalar multiplication (F x V → V). In contrast, a ring is a structure defined on a single set (R) with two internal operations, typically called addition (R x R → R) and multiplication (R x R → R). Essentially, a vector space involves an 'external' operation (scaling by a scalar), while a ring's operations are all 'internal' to its single set.
