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Vector Space Axiom Explorer

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About This Infographic

This interactive infographic helps students learn and remember the ten vector space axioms. The axioms are organized into two groups: five for vector addition and five for scalar multiplication. Click on each axiom card to see its definition and a concrete example.

Learning Objective: Students will identify and recognize the ten vector space axioms through an interactive concept map with hover definitions and example demonstrations.

How to Use

  1. Hover over the central hub to see what a vector space is and examples of vector spaces and fields
  2. Click on axiom cards to expand and see:
  3. Full definition of the axiom
  4. A concrete numerical example in R²
  5. Track your progress with the counter at the bottom showing how many axioms you've viewed
  6. A checkmark appears on viewed axioms

The Ten Vector Space Axioms

For a set V to be a vector space over a field F, it must satisfy:

Addition Axioms (1-5)

  1. Closure: u + v ∈ V
  2. Commutativity: u + v = v + u
  3. Associativity: (u + v) + w = u + (v + w)
  4. Identity: v + 0 = v
  5. Inverse: v + (-v) = 0

Scalar Multiplication Axioms (6-10)

  1. Closure: cv ∈ V
  2. Distributivity (vectors): c(u + v) = cu + cv
  3. Distributivity (scalars): (c + d)v = cv + dv
  4. Associativity: c(dv) = (cd)v
  5. Identity: 1·v = v

Why These Axioms Matter

These ten axioms are the foundation of linear algebra. Any set that satisfies all ten axioms is a vector space, which means all theorems about vector spaces apply to it. This includes:

  • R^n (n-dimensional real space)
  • Polynomials of degree ≤ n
  • Matrices of a given size
  • Continuous functions on an interval
  • Solutions to homogeneous differential equations

Embedding This Infographic

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<iframe src="https://dmccreary.github.io/linear-algebra/sims/vector-space-axiom-explorer/main.html"
        height="750px"
        width="100%"
        scrolling="no">
</iframe>

Lesson Plan

Grade Level

Undergraduate introductory linear algebra

Duration

15-20 minutes

Prerequisites

  • Vector addition and scalar multiplication concepts
  • Familiarity with R² as an example vector space

Learning Activities

  1. Introduction (3 min):
  2. Hover over the central hub to understand what a vector space is

  3. Note the two types of operations: addition and scalar multiplication

  4. Addition Axioms (5 min):

  5. Click through all five addition axioms
  6. Work through each example mentally or on paper

  7. Note how they formalize intuitive properties

  8. Scalar Multiplication Axioms (5 min):

  9. Click through all five scalar multiplication axioms
  10. Compare distributivity axioms 7 and 8

  11. Understand why the identity axiom uses 1

  12. Verification Exercise (5 min):

  13. Given a candidate vector space (e.g., 2×2 matrices)
  14. Check that all ten axioms hold

Discussion Questions

  1. Why do we need all ten axioms? What would go wrong if one was missing?
  2. Can you think of a set with addition that violates one of these axioms?
  3. Why is the additive identity (zero vector) unique?
  4. What's the difference between the two distributivity axioms?

Assessment Ideas

  • List all ten axioms from memory
  • Given a set and operations, verify which axioms hold
  • Explain why a given set is NOT a vector space

References

  1. 3Blue1Brown - Abstract vector spaces
  2. Khan Academy - Vector Spaces
  3. Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Section 3.1.
  4. Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Chapter 1.