Abstracting Vectors
Unit 3 treated vectors as arrows and tuples. Linear algebra abstracts away the picture and keeps only the operations. A vector space is any set of objects equipped with two operations, addition and scalar multiplication, satisfying a list of axioms. Anything obeying the axioms is a vector space.
The axioms formalize familiar rules:
| Property | Rule | Description |
|---|---|---|
| Addition | Commutative & associative | Order and grouping do not matter |
| Zero vector | Leaves vectors unchanged | |
| Inverse | Yields the zero vector | |
| Scaling | Distributes and scales properly |
Common pitfall: Thinking a vector must be an arrow or a tuple of numbers. In linear algebra, a vector is any element of a set satisfying the axioms. What matters is that you can add and scale them consistently.
Examples and Power
The power of abstraction is that many non-arrow objects satisfy the axioms and are true vector spaces:
| Space Type | Elements | Addition & Scaling |
|---|---|---|
| Familiar tuples | Coordinate-wise | |
| Polynomials | Degree | Combined by like powers |
| Matrices | Fixed size | Added and scaled entry-wise |
| Functions | Added and scaled pointwise | |
| Solutions | Differential equations | Homogeneous linear solutions |
Because all these share the same axioms, every theorem proved for abstract vector spaces applies to all of them at once. Proving a property for abstract vectors simultaneously establishes it for tuples, polynomials, matrices, and functions without repeating work.