Vector spaces are the foundation of linear algebra, defining sets of objects that behave like vectors. They're governed by axioms that dictate how vectors interact through addition and scalar multiplication. These rules ensure consistency and allow for powerful mathematical operations.

Understanding vector spaces is crucial for grasping more complex concepts in linear algebra. By mastering the axioms and properties, you'll be able to identify and work with various vector spaces, from simple number systems to abstract function spaces.

Vector spaces and their properties

Definition and essential components

• A vector space is a mathematical structure consisting of a set of elements called vectors, along with two operations: vector addition and scalar multiplication
• Vector addition takes two vectors and produces a new vector in the same set
  • Satisfies properties such as commutativity, associativity, and the existence of an identity element (zero vector)
• Scalar multiplication takes a scalar (typically a real or complex number) and a vector, producing a new vector in the same set
  • Satisfies properties such as distributivity and compatibility with scalar multiplication
• A vector space must have a unique zero vector, which when added to any vector, results in the same vector
• Every vector in a vector space must have an additive inverse, also called the negative of the vector
  • When the additive inverse is added to the original vector, the result is the zero vector

Additional properties and structures

• Different vector spaces may have additional properties, such as the presence of an inner product (inner product spaces) or a norm (normed vector spaces)
  • Inner product spaces allow for the definition of concepts like length, distance, and orthogonality between vectors
  • Normed vector spaces assign a non-negative real number (norm) to each vector, representing its magnitude or length
• Some vector spaces may have a basis, which is a linearly independent subset that spans the entire vector space
  • Bases allow for the representation of any vector in the space as a unique linear combination of basis vectors
• Vector spaces can be finite-dimensional or infinite-dimensional, depending on the cardinality of their basis (if it exists)
  • Finite-dimensional vector spaces have a basis with a finite number of elements
  • Infinite-dimensional vector spaces have a basis with an infinite number of elements or no basis at all

Axioms of vector spaces

Vector addition axioms

• Commutativity: $u + v = v + u$ for all vectors $u$ and $v$
• Associativity: $(u + v) + w = u + (v + w)$ for all vectors $u$, $v$, and $w$
• Identity element: There exists a unique zero vector $0$ such that $v + 0 = v$ for all vectors $v$
• Inverse element: For every vector $v$, there exists a unique vector $-v$ such that $v + (-v) = 0$

Scalar multiplication axioms

• Compatibility with scalar multiplication: $a(bv) = (ab)v$ for all scalars $a$, $b$, and vectors $v$
• Distributivity of scalar multiplication over vector addition: $a(u + v) = au + av$ for all scalars $a$ and vectors $u$, $v$
• Distributivity of scalar multiplication over scalar addition: $(a + b)v = av + bv$ for all scalars $a$, $b$, and vectors $v$
• Identity element for scalar multiplication: $1v = v$ for all vectors $v$, where $1$ is the multiplicative identity of the scalar field

Verifying vector spaces

Steps to verify a vector space

• To verify if a given set with two operations forms a vector space, one must check if all the axioms of a vector space are satisfied
• Begin by identifying the set of elements (vectors) and the two operations (vector addition and scalar multiplication)
• Check each axiom for vector addition: commutativity, associativity, existence of identity element, and existence of inverse elements
• Check each axiom for scalar multiplication: compatibility with scalar multiplication, distributivity over vector addition, distributivity over scalar addition, and the existence of the identity element for scalar multiplication
• If all axioms are satisfied, the given set with the two operations forms a vector space
  • If any axiom is not satisfied, the set is not a vector space

Common mistakes and pitfalls

• Forgetting to check all axioms, leading to incorrect conclusions about whether a set forms a vector space
• Confusing the properties of vector addition with those of scalar multiplication, or vice versa
• Not verifying the closure property, i.e., ensuring that the result of vector addition or scalar multiplication always yields a vector within the same set
• Misinterpreting the role of the zero vector or the scalar multiplicative identity in the axioms
• Failing to consider the underlying scalar field (e.g., real numbers, complex numbers) when verifying the axioms

Examples of vector spaces

Common vector spaces and their properties

• The set of all n-tuples of real numbers ($\mathbb{R}^n$) with component-wise addition and scalar multiplication is a vector space
  • Example: $\mathbb{R}^2$ represents the Euclidean plane, and $\mathbb{R}^3$ represents three-dimensional space
• The set of all $m \times n$ matrices with real entries ($M_{m,n}(\mathbb{R})$) forms a vector space under matrix addition and scalar multiplication
  • Example: $M_{2,2}(\mathbb{R})$ is the vector space of all $2 \times 2$ matrices with real entries
• The set of all polynomials with real coefficients ($P(\mathbb{R})$) is a vector space under polynomial addition and scalar multiplication
  • Example: $P_3(\mathbb{R})$ is the vector space of all polynomials with real coefficients and degree at most 3
• The set of all continuous functions from a closed interval $[a, b]$ to the real numbers ($C[a, b]$) forms a vector space under point-wise addition and scalar multiplication
  • Example: $C[0, 1]$ is the vector space of all continuous real-valued functions on the interval $[0, 1]$
• The set of all sequences of real numbers is a vector space under component-wise addition and scalar multiplication
  • Example: The space of all real-valued sequences $(a_n)_{n=1}^{\infty}$ is an infinite-dimensional vector space

Non-examples and their violations

• The set of all natural numbers $\mathbb{N}$ under addition and multiplication is not a vector space
  • Fails the axiom of inverse elements for vector addition, as there are no negative natural numbers
• The set of all non-zero real numbers $\mathbb{R} \setminus {0}$ under standard multiplication and addition is not a vector space
  • Fails the axiom of identity element for vector addition, as 0 is not included in the set
• The set of all positive real numbers $\mathbb{R}^+$ under standard addition and multiplication is not a vector space
  • Fails the axiom of inverse elements for vector addition, as the additive inverse of a positive number is negative
• The set of all real-valued functions on $\mathbb{R}$ under function composition and scalar multiplication is not a vector space
  • Function composition is not commutative, violating the commutativity axiom for vector addition