Vector Space
Definition
Discover the fundamental algebraic structure that unifies arrows, functions, and polynomials under one powerful framework.
Linear Algebra
What Is a Vector Space?
A vector space (also called a linear space) is a collection of objects called vectors that can be added together and scaled by numbers, and these operations behave in predictable, consistent ways.
The Two Operations
Vector Addition
Takes two vectors \( \mathbf{u}, \mathbf{v} \in V \) and produces a new vector \( \mathbf{u} + \mathbf{v} \in V \).
Scalar Multiplication
Takes a scalar \( c \in F \) and a vector \( \mathbf{v} \in V \) and produces a new vector \( c\,\mathbf{v} \in V \).
Visual Intuition in \( \mathbb{R}^2 \)
The Eight Axioms: Addition
1
Closure under addition: If \( \mathbf{u}, \mathbf{v} \in V \), then \( \mathbf{u} + \mathbf{v} \in V \).
2
Commutativity: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
3
Associativity: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
4
Zero vector: There exists \( \mathbf{0} \in V \) such that \( \mathbf{v} + \mathbf{0} = \mathbf{v} \) for all \( \mathbf{v} \).
5
Additive inverse: For each \( \mathbf{v} \), there exists \( -\mathbf{v} \) such that \( \mathbf{v} + (-\mathbf{v}) = \mathbf{0} \).
The Eight Axioms: Scalar Multiplication
6
Closure under scalar multiplication: If \( c \in F \) and \( \mathbf{v} \in V \), then \( c\,\mathbf{v} \in V \).
7
Distributivity (scalar over vector addition): \( c(\mathbf{u} + \mathbf{v}) = c\,\mathbf{u} + c\,\mathbf{v} \).
8
Distributivity (field over scalar multiplication): \( (a + b)\mathbf{v} = a\,\mathbf{v} + b\,\mathbf{v} \).
Formal Definition
Vector Space vs. Subspace
| Vector Space | Subspace | |
|---|---|---|
| Definition | Set \( V \) with two operations satisfying all 8 axioms | A subset \( W \subseteq V \) that is itself a vector space |
| Must verify | All 8 axioms independently | Only 3 conditions: nonempty, closed under \( + \), closed under \( c\cdot \) |
| Contains \( \mathbf{0} \) | By axiom 4 | Follows from closure (must check) |
| Example | \( \mathbb{R}^3 \) | A plane through the origin in \( \mathbb{R}^3 \) |
Classic Examples
\( \mathbb{R}^n \)
The set of all \( n \)-tuples of real numbers with component-wise addition and scalar multiplication.
\( P_n \)
Polynomials of degree \( \leq n \). Addition and scalar multiplication are defined term by term.
\( M_{m \times n} \)
All \( m \times n \) matrices over \( \mathbb{R} \). Addition is entry-wise, scaling multiplies every entry.
How to Verify a Vector Space
1
Identify the set and field: State \( V \) and the scalar field \( F \).
2
Define the operations: Write explicit rules for \( \mathbf{u}+\mathbf{v} \) and \( c\,\mathbf{v} \).
3
Check closure: Show both operations produce elements still in \( V \).
4
Check the remaining axioms: Commutativity, associativity, zero vector, additive inverse, distributivity, scalar associativity, identity.
5
Conclude: If all axioms hold, \( V \) is a vector space over \( F \).
Example 1: \( \mathbb{R}^2 \) Is a Vector Space
Set: \( V = \mathbb{R}^2 = \{(x,y) : x,y \in \mathbb{R}\} \). Field: \( F = \mathbb{R} \).
✓
Closure: Adding two pairs gives a pair; scaling a pair gives a pair. Both stay in \( \mathbb{R}^2 \).
✓
Zero vector: \( \mathbf{0} = (0,0) \). For any \( (x,y) \): \( (x,y)+(0,0) = (x,y) \).
✓
Additive inverse: \( -(x,y) = (-x,-y) \). All other axioms follow from real-number arithmetic.
Example 2: Polynomials \( P_2 \)
Set: \( P_2 = \{a_0 + a_1 t + a_2 t^2 : a_i \in \mathbb{R}\} \). Field: \( F = \mathbb{R} \).
✓
Closure: Sum of two degree-\( \leq 2\) polynomials has degree \( \leq 2\). Scaling preserves degree.
✓
Zero vector: The zero polynomial \( 0 + 0t + 0t^2 \). Adding it changes nothing.
✓
All axioms hold because the coefficients are real numbers, inheriting the field properties of \( \mathbb{R} \).
Example 3: A Non-Example
Consider the set \( S = \{(x,y) \in \mathbb{R}^2 : x \geq 0,\; y \geq 0\} \), the first quadrant, with usual addition and scalar multiplication.
Why Vector Spaces Matter
Unification
Arrows, matrices, polynomials, and functions all obey the same abstract rules.
Dimension
Once you know it is a vector space, you can find a basis and count its dimension.
Applications
Computer graphics, machine learning, quantum mechanics, and signal processing all rest on vector-space theory.
Common Mistakes
✕ Forgetting the Zero Vector
A set that does not contain a zero element cannot be a vector space. Always check axiom 4 first.
✕ Ignoring Scalar Closure
Checking addition closure is not enough. Scalar multiplication by negative or fractional scalars can push vectors outside the set.
✕ Custom Operations Trap
When operations are defined unusually (e.g., \( x \oplus y = xy \)), you must re-verify every axiom: nothing can be assumed.
Question 1 of 5
Which of the following is not one of the vector space axioms?
Vector spaces require additive inverses, not multiplicative ones. There is no general notion of "dividing" one vector by another in a vector space.
Question 2 of 5
Which set with standard operations is a vector space over \( \mathbb{R} \)?
The set \( M_{2\times 2}(\mathbb{R}) \) with entry-wise addition and scalar multiplication satisfies all axioms. The other sets fail closure: e.g., scaling a determinant-1 matrix by 2 gives determinant 4.
Question 3 of 5
The set \( S = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = 1\} \) with standard operations is:
Since \( 0+0+0 = 0 \neq 1 \), the zero vector is not in \( S \). Without a zero vector, axiom 4 fails immediately. (Addition closure also fails: \( (1,0,0)+(0,1,0)=(1,1,0) \) and \( 1+1+0=2 \neq 1 \).)
Question 4 of 5
In the vector space \( P_2 \) of polynomials of degree \( \leq 2 \), what is the zero vector?
The zero vector in a polynomial space is the polynomial whose every coefficient is zero. Adding it to any polynomial \( p(t) \) gives \( p(t) \) back, satisfying axiom 4.
Question 5 of 5
To show a subset \( W \) of a known vector space \( V \) is a subspace, how many conditions must you verify?
Because \( W \) inherits commutativity, associativity, distributivity, etc., from \( V \), you only need to verify the three closure-type conditions: the zero vector is in \( W \), addition stays in \( W \), and scalar multiplication stays in \( W \).