Re-imagining Matrices

2025.11.29

Linear algebra feels abstract until you realize it behaves exactly like a kitchen.

A matrix is a pantry.
A vector is a recipe.
Matrix multiplication is simply cooking.

This single metaphor reproduces all three classical views—column combinations, systems, and transformations—without distortion.

1. Matrices as Pantries

Write a matrix as

A = [a_1\; a_2\; \dots\; a_n].

Each column (a_i) is an ingredient with a fixed flavor profile.
Salt, chili, lemon—encoded as vectors.

2. Vectors as Recipes

A vector

x = (x_1, x_2, \dots, x_n)

is the recipe: the amount of each ingredient you choose.

Nothing mystical. Just quantities.

3. Cooking: Linear Combination

The dish produced by the kitchen is

Ax = x_1 a_1 + x_2 a_2 + \cdots + x_n a_n.

This is the column-combination view.
Every dish is a weighted mixture of base ingredients.

4. Systems: Matching a Target Dish

Given a target flavor profile (b), solving

Ax = b

means: find a recipe whose cooked dish hits that profile exactly.

Row operations are simply constraints on taste—saltiness here, acidity there.

5. Null Space: Flavor Cancellation

The null space

\ker(A) = \{x : Ax = 0\}

contains all recipes whose ingredients cancel to a neutral dish.
These directions reflect redundancy in the pantry: flavors that undo one another.

6. Eigenvectors: Stable Flavor Notes

A vector (v) is an eigenvector when

Av = \lambda v.

These are the recipes whose fundamental flavor survives the cooking process.
The transformation scales the intensity but preserves the taste.

7. The Operator View

Interpreting (A) as a linear transformation is simply reframing the process:

input: the flavor profile of a raw dish
output: the transformed flavor after it passes through the kitchen

A linear map is a kitchen whose behavior is fully determined by how it cooks the basis ingredients.

This is the entire subject reduced to one metaphor:
ingredients, recipes, dishes.
Everything else—rank, projections, diagonalization—follows from this.