Vector Space Properties

Why This Matters

Vector space properties aren't just abstract rules to memorize—they're the foundation of linear algebra itself. Every theorem you'll encounter about linear transformations, bases, dimension, and eigenspaces depends on these ten axioms holding true. When you're asked to prove something is a vector space or identify why a particular set fails to be one, you're being tested on your understanding of closure, identity elements, inverses, and distributive laws.

Think of these properties as a checklist that separates genuine vector spaces from imposters. The exam loves to give you a set with some operations and ask "Is this a vector space?"—and the answer always comes down to which axiom breaks. Don't just memorize the list; know what each property guarantees and what goes wrong when it fails. That's where the points are.

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Closure Properties

These two axioms ensure that performing operations on vectors in your space doesn't accidentally kick you outside the space. If a set isn't closed, it can't be a vector space—full stop.

Closure Under Addition

• The sum of any two vectors stays in the space—if $u, v \in V$, then $u + v \in V$
• This is the first thing to check when verifying a subspace; many "fake" vector spaces fail here
• Geometric intuition: the space contains all "combinations" of its elements, not just isolated points

Closure Under Scalar Multiplication

• Scaling a vector by any scalar keeps it in the space—if $v \in V$ and $c$ is a scalar, then $c \cdot v \in V$
• The scalar field matters: for real vector spaces, $c \in \mathbb{R}$; for complex, $c \in \mathbb{C}$
• Common failure point: sets like "all vectors with integer components" fail this axiom over $\mathbb{R}$

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Additive Structure Properties

These four axioms govern how vector addition behaves. Together with the closure axiom, they make $(V, +)$ an abelian group—a structure you'll see throughout abstract algebra.

Associativity of Addition

• Grouping doesn't matter: $(u + v) + w = u + (v + w)$ for all $u, v, w \in V$
• Practical payoff: you can write $u + v + w$ without parentheses and compute in any order
• This axiom rarely fails in naturally-defined spaces but is essential for proving general theorems

Commutativity of Addition

• Order doesn't matter: $u + v = v + u$ for all $u, v \in V$
• This makes vector addition symmetric—unlike matrix multiplication, which isn't commutative
• Key distinction: vector spaces require commutative addition; modules over non-commutative rings don't

Existence of Zero Vector

• Every vector space has an additive identity $\mathbf{0}$ such that $v + \mathbf{0} = v$ for all $v \in V$
• The zero vector is unique—you can prove this using the axioms themselves
• Subspace test essential: if a subset doesn't contain $\mathbf{0}$, it's automatically not a subspace

Existence of Additive Inverse

• Every vector has a "negative": for each $v \in V$, there exists $-v$ such that $v + (-v) = \mathbf{0}$
• This defines subtraction: $u - v$ means $u + (-v)$
• Uniqueness follows from the axioms—a favorite short proof on exams

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Scalar Multiplication Properties

These four axioms connect the scalar field to the vector space. They ensure that scalars interact "nicely" with vectors and with each other—distributivity is where the two operations truly link up.

Distributivity Over Vector Addition

• Scalars distribute across vector sums: $c \cdot (u + v) = c \cdot u + c \cdot v$
• This connects scalar multiplication to the additive structure of the vector space
• Proof tool: essential for showing that scalar multiples of subspaces remain subspaces

Distributivity Over Scalar Addition

• Scalar sums distribute across vectors: $(a + b) \cdot v = a \cdot v + b \cdot v$
• This connects vector space structure to the field structure of the scalars
• Often confused with the previous axiom—note which addition is being distributed over

Associativity of Scalar Multiplication

• Scalars compose naturally: $(ab) \cdot v = a \cdot (b \cdot v)$ for scalars $a, b$ and vector $v$
• Not listed in original but implied—this axiom ensures scalar multiplication is compatible with field multiplication
• Rarely fails in standard examples but critical for the formal definition

Scalar Multiplication Identity

• Multiplying by 1 does nothing: $1 \cdot v = v$ for all $v \in V$
• The "1" comes from the scalar field—it's the multiplicative identity of $\mathbb{R}$ or $\mathbb{C}$
• Without this axiom, scalar multiplication could collapse everything to zero

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Subspace Verification

This isn't an axiom but a derived criterion that packages the axioms into a practical test. Mastering subspace verification is one of the most exam-relevant skills in this course.

Subspace Criteria

• Three-part test: a subset $W \subseteq V$ is a subspace if it contains $\mathbf{0}$, is closed under addition, and is closed under scalar multiplication
• Shortcut version: check that $W$ is closed under linear combinations—if $u, v \in W$ and $a, b$ are scalars, then $au + bv \in W$
• The other axioms are inherited from the parent space $V$, so you don't re-verify associativity, commutativity, etc.

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Quick Reference Table

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Self-Check Questions

1. A subset $W$ of $\mathbb{R}^3$ contains the vectors $(1, 0, 0)$ and $(0, 1, 0)$ but not $(1, 1, 0)$. Which axiom does $W$ violate, and why can't it be a subspace?

2. Compare the two distributivity axioms. Write out both symbolically and explain what "thing" is being distributed in each case.

3. If someone defines a "vector space" where $1 \cdot v = \mathbf{0}$ for all $v$, which axiom fails? What would go wrong with the structure?

4. Why don't you need to verify associativity of addition when checking whether a subset is a subspace? Which properties do you need to verify?

5. The set of all polynomials of degree exactly 2 (like $x^2 + 3x + 1$) is not a subspace of all polynomials. Identify two different axioms this set violates and give a specific example for each.