The Anatomy of Vector Spaces
A vector is a list of numbers representing data like asset weights or sales. A vector space is the universe built from vectors using two moves: addition and scaling.
Linear combination: the core building move, written as . Mixing stocks and bonds in any proportions creates a linear combination. The set of all such combinations is their span, the entire reachable territory.
| Concept | Definition | Practical Test |
|---|---|---|
| Span | All reachable linear combinations | Territory you can build |
| Linear Independence | No vector is a combination of others | No redundancy |
| Basis | Independent and spanning set | Minimal coordinate system |
Linear independence means no vector is redundant. If fund C is a 50/50 split of funds A and B, adding it spans nothing new.
Basis, Dimension, and Variety
A basis is a set that is both independent (no waste) and spanning (no gaps). Dimension is the number of vectors in any basis, counting genuine degrees of freedom.
| Property | Meaning |
|---|---|
| Dimension | Count of independent directions, not listed items |
| Matrix Rank | Number of independent vectors when stacked as columns |
| Determinant | for independent vectors in |
Common pitfall: Confusing the number of listed items with the true dimension. Three portfolios built from just two funds still span a 2D space. Diversifying across dependent assets is buying identical risk in different wrappers.
Tip: To test independence, stack vectors as columns and compute the rank. Rank equals the vector count if and only if they are independent.