What are vector spaces under multiplication?
The concept of vector spaces under multiplication is a common source of confusion because vector spaces are not defined under multiplication of vectors. Let me clarify:
๐ What Is a Vector Space?
A vector space over a field (like the real numbers โ) is a set equipped with:
- Vector addition: Combining two vectors to get another vector.
- Scalar multiplication: Multiplying a vector by a scalar (from the field) to get another vector.
These operations must satisfy eight axioms (closure, associativity, identity elements, distributivity, etc.).
❌ Multiplication of Vectors? Not a Vector Space Operation
- Vector multiplication (like dot product, cross product, or element-wise multiplication) is not part of the vector space definition.
- These products don't satisfy the vector space axioms when used as the main operation.
For example:
- The dot product takes two vectors and gives a scalar — not another vector.
- The cross product only exists in โ³ and isn't associative or defined for general vector spaces.
- Component-wise multiplication doesn't preserve closure in all cases and doesn't satisfy vector space axioms.
✅ Algebraic Structures with Multiplication
If you're thinking about combining vector-like objects with multiplication, you're likely looking at one of these more advanced structures:
| Structure Type | Description |
|---|---|
| Algebra | A vector space with a bilinear multiplication operation (e.g., matrix algebra). |
| Ring | A set with two operations (addition and multiplication) but not necessarily scalar multiplication. |
| Field | A ring where every nonzero element has a multiplicative inverse. |
| Module | A generalization of vector spaces over rings instead of fields. |
๐ Example: Matrix Algebra
Matrices form a vector space under addition and scalar multiplication, and also support multiplication (matrix multiplication), which makes them an algebra.
Summary
There is no such thing as a "vector space under multiplication" if you mean multiplication of vectors. Vector spaces are defined with addition and scalar multiplication only. If you're exploring structures where multiplication of elements is involved, you're likely referring to algebras or modules, not vector spaces in the strict sense.
Let me know if you meant something more specific!
No comments:
Post a Comment