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What is a vector space and examples

A vector space (or linear space) is a fundamental concept in mathematics, specifically in linear algebra. It is a set of objects called vectors, along with two operations: vector addition and scalar multiplication, that satisfy a specific set of rules.

Formal Definition:

A vector space V over a field F (like the real numbers \mathbb{R} or complex numbers \mathbb{C}) is a set equipped with two operations:

  1. Vector Addition: Combines two vectors \mathbf{u}, \mathbf{v} \in V to produce another vector \mathbf{u} + \mathbf{v} \in V.
  2. Scalar Multiplication: Multiplies a scalar c \in F with a vector \mathbf{v} \in V to produce a new vector c\mathbf{v} \in V.

These operations must satisfy the following axioms:

Axioms of a Vector Space:

  1. Closure under addition and scalar multiplication:
  • If \mathbf{u}, \mathbf{v} \in V, then \mathbf{u} + \mathbf{v} \in V.
  • If c \in F and \mathbf{v} \in V, then c\mathbf{v} \in V.
  1. Associativity of addition: (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) for all \mathbf{u}, \mathbf{v}, \mathbf{w} \in V.
  2. Commutativity of addition: \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} for all \mathbf{u}, \mathbf{v} \in V.
  3. Additive identity: There exists a vector \mathbf{0} \in V such that \mathbf{v} + \mathbf{0} = \mathbf{v} for all \mathbf{v} \in V.
  4. Additive inverse: For each \mathbf{v} \in V, there exists -\mathbf{v} \in V such that \mathbf{v} + (-\mathbf{v}) = \mathbf{0}.
  5. Distributive properties:
  • c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v} for all c \in F and \mathbf{u}, \mathbf{v} \in V.
  • (c + d)\mathbf{v} = c\mathbf{v} + d\mathbf{v} for all c, d \in F and \mathbf{v} \in V.
  1. Associativity of scalar multiplication: c(d\mathbf{v}) = (cd)\mathbf{v} for all c, d \in F and \mathbf{v} \in V.
  2. Identity scalar multiplication: 1\mathbf{v} = \mathbf{v} for all \mathbf{v} \in V, where 1 is the multiplicative identity in F.

Examples of Vector Spaces:

  1. Real coordinate spaces:
  • The set \mathbb{R}^n, where vectors are n-tuples of real numbers (e.g., \mathbf{v} = (v_1, v_2, \dots, v_n)), is a vector space over \mathbb{R}.
  1. Polynomials:
  • The set of all polynomials with real coefficients forms a vector space over \mathbb{R}.
  1. Matrices:
  • The set of all m \times n matrices is a vector space over \mathbb{R} or \mathbb{C}.
  1. Functions:
  • The set of all real-valued functions is a vector space over \mathbb{R}.

Importance:

Vector spaces provide the framework for many areas of mathematics, physics, and engineering. They underpin concepts like systems of linear equations, transformations, and more advanced fields like quantum mechanics and machine learning.

Prove that the real coordinate space is a vector space

To verify that the real coordinate space \mathbb{R}^n is a vector space, we must show that it satisfies the axioms of a vector space.

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Let \mathbb{R}^n represent the set of all n-tuples of real numbers, where a vector \mathbf{v} \in \mathbb{R}^n is written as:

\mathbf{v} = (v_1, v_2, \dots, v_n), \quad v_i \in \mathbb{R}.

We define:

  1. Vector addition: For \mathbf{u}, \mathbf{v} \in \mathbb{R}^n,
    \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, \dots, u_n + v_n).
  2. Scalar multiplication: For c \in \mathbb{R} and \mathbf{v} \in \mathbb{R}^n,
    c \mathbf{v} = (c v_1, c v_2, \dots, c v_n).

We now check the vector space axioms.

1. Closure under addition:

If \mathbf{u} = (u_1, u_2, \dots, u_n) and \mathbf{v} = (v_1, v_2, \dots, v_n), then:
\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, \dots, u_n + v_n).
Since u_i + v_i \in \mathbb{R} for all i, \mathbf{u} + \mathbf{v} \in \mathbb{R}^n. (Satisfied)

2. Closure under scalar multiplication:

If c \in \mathbb{R} and \mathbf{v} = (v_1, v_2, \dots, v_n), then:
c\mathbf{v} = (c v_1, c v_2, \dots, c v_n).
Since c v_i \in \mathbb{R} for all i, c\mathbf{v} \in \mathbb{R}^n. (Satisfied)

3. Associativity of addition:

For \mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n,
(\mathbf{u} + \mathbf{v}) + \mathbf{w} = ((u_1 + v_1) + w_1, \dots, (u_n + v_n) + w_n),
\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (u_1 + (v_1 + w_1), \dots, u_n + (v_n + w_n)).
Since addition in \mathbb{R} is associative, (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}). (Satisfied)

4. Commutativity of addition:

For \mathbf{u}, \mathbf{v} \in \mathbb{R}^n,
\mathbf{u} + \mathbf{v} = (u_1 + v_1, \dots, u_n + v_n) = (v_1 + u_1, \dots, v_n + u_n) = \mathbf{v} + \mathbf{u}.
Since addition in \mathbb{R} is commutative, this holds. (Satisfied)

5. Additive identity:

The zero vector \mathbf{0} = (0, 0, \dots, 0) \in \mathbb{R}^n satisfies:
\mathbf{v} + \mathbf{0} = (v_1 + 0, v_2 + 0, \dots, v_n + 0) = (v_1, v_2, \dots, v_n) = \mathbf{v}.
(Satisfied)

6. Additive inverse:

For \mathbf{v} = (v_1, v_2, \dots, v_n), the additive inverse is -\mathbf{v} = (-v_1, -v_2, \dots, -v_n), since:
\mathbf{v} + (-\mathbf{v}) = (v_1 - v_1, v_2 - v_2, \dots, v_n - v_n) = (0, 0, \dots, 0) = \mathbf{0}.
(Satisfied)

7. Distributive properties:

For c, d \in \mathbb{R} and \mathbf{v}, \mathbf{u} \in \mathbb{R}^n:

  • c(\mathbf{u} + \mathbf{v}) = c(u_1 + v_1, \dots, u_n + v_n) = (c u_1 + c v_1, \dots, c u_n + c v_n) = c\mathbf{u} + c\mathbf{v}.
  • (c + d)\mathbf{v} = ((c + d)v_1, \dots, (c + d)v_n) = (cv_1 + dv_1, \dots, cv_n + dv_n) = c\mathbf{v} + d\mathbf{v}.
    (Satisfied)

8. Associativity of scalar multiplication:

For c, d \in \mathbb{R} and \mathbf{v} \in \mathbb{R}^n,
c(d\mathbf{v}) = c(d v_1, \dots, d v_n) = ((cd)v_1, \dots, (cd)v_n) = (cd)\mathbf{v}.
(Satisfied)

9. Identity scalar multiplication:

For \mathbf{v} \in \mathbb{R}^n,
1\mathbf{v} = (1 v_1, 1 v_2, \dots, 1 v_n) = (v_1, v_2, \dots, v_n) = \mathbf{v}.
(Satisfied)

Conclusion:

Since \mathbb{R}^n satisfies all the axioms of a vector space, it is indeed a vector space.

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