Vector Space Axiom Explorer
Click axioms to explore definitions and examples
Vector Space V over Field F
Vector Space: A set V of objects (vectors) with two operations (addition and scalar multiplication) satisfying 10 axioms.
Examples of V: R², R³, polynomials, matrices, functions
Examples of F: Real numbers (R), Complex numbers (C)
Examples of V: R², R³, polynomials, matrices, functions
Examples of F: Real numbers (R), Complex numbers (C)
Addition Axioms (5)
1. Closure under Addition
u + v ∈ V
For any two vectors u and v in V, their sum u + v is also in V.
The space is "closed" under addition.
Example in R²:
(1, 2) + (3, 4) = (4, 6) ∈ R²
(1, 2) + (3, 4) = (4, 6) ∈ R²
2. Commutativity
u + v = v + u
The order of addition doesn't matter. You can add vectors in any order and get the same result.
Example:
(1, 2) + (3, 4) = (3, 4) + (1, 2) = (4, 6)
(1, 2) + (3, 4) = (3, 4) + (1, 2) = (4, 6)
3. Associativity
(u + v) + w = u + (v + w)
Grouping doesn't matter when adding three or more vectors.
You can add them in any grouping.
Example:
((1,0) + (0,1)) + (1,1) = (1,0) + ((0,1) + (1,1))
= (2, 2)
((1,0) + (0,1)) + (1,1) = (1,0) + ((0,1) + (1,1))
= (2, 2)
4. Additive Identity
v + 0 = v
There exists a zero vector 0 such that adding it to any vector v leaves v unchanged.
Example in R²:
(3, 4) + (0, 0) = (3, 4)
The zero vector is (0, 0).
(3, 4) + (0, 0) = (3, 4)
The zero vector is (0, 0).
5. Additive Inverse
v + (-v) = 0
Every vector v has an opposite -v such that their sum is the zero vector.
Example:
(3, 4) + (-3, -4) = (0, 0)
The inverse of (3, 4) is (-3, -4).
(3, 4) + (-3, -4) = (0, 0)
The inverse of (3, 4) is (-3, -4).
Scalar Multiplication Axioms (5)
6. Closure under Scalar Multiplication
cv ∈ V
For any scalar c and vector v in V, the product cv is also in V.
Scaling a vector keeps it in the space.
Example:
3 · (1, 2) = (3, 6) ∈ R²
3 · (1, 2) = (3, 6) ∈ R²
7. Distributivity (Vectors)
c(u + v) = cu + cv
A scalar multiplied by a sum of vectors equals the sum of the scalar times each vector.
Example:
2·((1,0) + (0,1)) = 2·(1,0) + 2·(0,1)
2·(1,1) = (2,0) + (0,2) = (2, 2) ✓
2·((1,0) + (0,1)) = 2·(1,0) + 2·(0,1)
2·(1,1) = (2,0) + (0,2) = (2, 2) ✓
8. Distributivity (Scalars)
(c + d)v = cv + dv
A sum of scalars times a vector equals the sum of each scalar times the vector.
Example:
(2 + 3)·(1, 2) = 2·(1, 2) + 3·(1, 2)
5·(1, 2) = (2, 4) + (3, 6) = (5, 10) ✓
(2 + 3)·(1, 2) = 2·(1, 2) + 3·(1, 2)
5·(1, 2) = (2, 4) + (3, 6) = (5, 10) ✓
9. Associativity (Scalars)
c(dv) = (cd)v
Scaling by c then d is the same as scaling by their product cd.
Example:
2·(3·(1, 2)) = (2·3)·(1, 2)
2·(3, 6) = 6·(1, 2) = (6, 12) ✓
2·(3·(1, 2)) = (2·3)·(1, 2)
2·(3, 6) = 6·(1, 2) = (6, 12) ✓
10. Scalar Identity
1 · v = v
Multiplying any vector by the scalar 1 leaves the vector unchanged.
Example:
1 · (3, 4) = (3, 4)
The identity scalar is 1.
1 · (3, 4) = (3, 4)
The identity scalar is 1.