Vector Space Definition Axioms Properties and Solved Examples
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 in Linear Algebra Explained Clearly
1. What is a vector space in linear algebra?
A vector space is a set of vectors equipped with vector addition and scalar multiplication that satisfy specific axioms. A set V over a field F (such as ℝ or ℂ) is a vector space if it satisfies properties like:
- Closure under addition and scalar multiplication
- Commutativity and associativity of addition
- Existence of a zero vector
- Existence of an additive inverse for each vector
- Distributive and scalar multiplication laws
2. What are the axioms of a vector space?
The axioms of a vector space are rules that define how vectors behave under addition and scalar multiplication. For vectors u, v, w in V and scalars a, b in F:
- u + v = v + u (commutativity)
- (u + v) + w = u + (v + w) (associativity)
- There exists a zero vector 0 such that u + 0 = u
- For each u, there exists −u such that u + (−u) = 0
- a(u + v) = au + av (distributive law)
- (a + b)u = au + bu
- a(bu) = (ab)u
- 1u = u
3. Can you give an example of a vector space?
An example of a vector space is ℝ², the set of all ordered pairs (x, y) of real numbers. In ℝ²:
- Vector addition: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)
- Scalar multiplication: a(x, y) = (ax, ay)
4. What is a subspace of a vector space?
A subspace is a subset of a vector space that is itself a vector space under the same operations. A subset W of V is a subspace if:
- It contains the zero vector
- It is closed under vector addition
- It is closed under scalar multiplication
5. What is the dimension of a vector space?
The dimension of a vector space is the number of vectors in its basis. A basis is a set of linearly independent vectors that span the space. For example:
- dim(ℝ²) = 2
- dim(ℝ³) = 3
- The space of polynomials of degree ≤ 2 has dimension 3
6. What is a basis of a vector space?
A basis of a vector space is a set of vectors that are linearly independent and span the space. This means:
- No vector in the set can be written as a linear combination of others
- Every vector in the space can be written as a linear combination of the basis vectors
7. What does it mean for vectors to be linearly independent?
Vectors are linearly independent if the only solution to a₁v₁ + a₂v₂ + … + aₙvₙ = 0 is a₁ = a₂ = … = aₙ = 0. This means none of the vectors can be expressed as a combination of the others. For example, in ℝ², (1,0) and (0,1) are linearly independent, but (1,2) and (2,4) are linearly dependent because (2,4) = 2(1,2).
8. How do you check if a set is a vector space?
To check if a set is a vector space, verify that it satisfies all vector space axioms. The key steps are:
- Confirm closure under addition and scalar multiplication
- Check the existence of a zero vector
- Verify existence of additive inverses
- Ensure distributive and associative laws hold
9. What is the zero vector in a vector space?
The zero vector is the unique vector 0 in a vector space such that v + 0 = v for every vector v. In ℝ², the zero vector is (0,0), and in ℝ³ it is (0,0,0). The zero vector acts as the additive identity element in the space.
10. What is the difference between a vector space and a subspace?
A vector space is the entire set that satisfies the vector space axioms, while a subspace is a subset that also satisfies those same axioms. The difference is:
- A vector space stands on its own
- A subspace must be contained within a larger vector space
- Every subspace must include the zero vector of the parent space
