🧠 Approved Operations
Linear algebra is the mathematics of vectors, matrices, and linear transformations. It governs systems where scaling and adding behave predictably—preserving structure, direction, and proportionality.
It’s not just “math on arrays”—it’s the logic of linear relationships, used in geometry, physics, computer graphics, machine learning, and more.
🔹 Core Objects
| Object | Description |
|---|---|
| Scalars | Single numbers (from a field like $\mathbb{R}$ or $\mathbb{C}$) |
| Vectors | Ordered lists of scalars (1D arrays) |
| Matrices | Rectangular arrays of scalars (2D arrays) |
| Linear maps | Functions that preserve addition and scalar multiplication |
🔹 Approved Operations in Linear Algebra
| Operation | Description | Preserves Linearity? |
|---|---|---|
| Vector addition | $x + y$ | ✅ |
| Scalar multiplication | $a \cdot x$ | ✅ |
| Matrix addition | $A + B$ | ✅ |
| Matrix multiplication | $AB$ | ✅ |
| Dot product | $x \cdot y$ | ✅ |
| Transpose | $A^T$ | ✅ |
| Inverse (when defined) | $A^{-1}$ | ✅ |
| Identity matrix | $I_n$ | ✅ |
| Zero matrix | $0_{n \times m}$ | ✅ |
These operations are structure-preserving and support solving systems, composing transformations, and modeling geometry.
🔹 Why Some Operations Are Not Approved
Linear algebra excludes operations that break linearity, lack transformation logic, or don’t generalize.
| Operation | Description | Reason for Exclusion |
|---|---|---|
| Hadamard product | Element-wise multiplication | ❌ Breaks linearity |
| Cross product | Only defined in $\mathbb{R}^3$ | ❌ Not generalizable |
| ReLU / activation | Nonlinear functions | ❌ Breaks linearity |
| Vector multiplication | Ambiguous (no canonical definition) | ❌ Violates structure |
| Entry-wise division | Undefined in matrix algebra | ❌ No transformation meaning |
🔹 Semantic Audit: What Linear Algebra Requires
| Requirement | Why It Matters |
|---|---|
| Linearity | Ensures predictable behavior under scaling and addition |
| Dimensional consistency | Guarantees valid compositions |
| Associativity | Enables chaining of transformations |
| Closure | Keeps results inside the same space |
| Generalizability | Works across $\mathbb{R}^n$, $\mathbb{C}^n$, etc. |
🔹 Philosophical Framing
Linear algebra is the language of linear structure:
- It models how systems respond to scaling and combination
- It encodes geometry, symmetry, and composition
- It excludes operations that don’t respect these principles
Would you like to scaffold a companion module on nonlinear operations and how they extend or break the linear framework—especially in neural nets, optimization, or physics? It’s a great contrast for vault clarity.