Tea Bags and Linear Algebra

When we brew a cup of tea, we rarely think about mathematics. However, the relationship between the number of tea bags we use and the volume of water required provides a perfect case study for the limits of linear transformations. At first, the process looks linear—double the tea, double the water—but the rigorous mathematical definition of a linear transformation requires more than just a simple proportion.

The Problem of Domain and Codomain #

In the mathematical sense, a function \( f: T \to V \) mapping a set of tea bags \( T \) to a set of water volumes \( V \) is not a linear transformation. For a function to be considered a linear transformation, both its domain and codomain must be vector spaces over the same field.

The set of physical tea bags \( T \) fails this requirement because it lacks the necessary algebraic structure. A vector space requires a well-defined operation of addition where the sum of two elements remains in the set. Adding one Earl Grey tea bag to one Green tea bag does not result in a single tea bag element within a uniform set; it results in a collection of two tea bags.
Furthermore, a vector space must contain a zero element (the additive identity) and additive inverses. Since negative tea bags do not exist in physical reality, the set \( T \) cannot be a vector space.

Transitioning to Mathematical Modeling #

To make the problem mathematically tractable, we shift our focus from physical objects to their measures. If we define \( x \) as the number of tea bags, the domain becomes a subset of the positive real numbers, \( \mathbb{R}_{\ge 0} \). This allows us to define a function based on the desired concentration of the infusion.

The choice of the expression \( f(x) = kx \) is justified by two primary factors: physical constraints and analytical requirements. Physically, the taste of tea depends on the concentration \( C \), which is the ratio of tea bags \( x \) to the volume of water \( V \):

$$C = \frac{x}{V}$$

To maintain a consistent flavor, \( C \) must be a constant \( C_{0} \) (e.g. 1 tea bag per 250ml of water). Solving for \( V \) gives \( V = \left(\frac{1}{C_{0}}\right) \cdot x \).
By setting \( k = 1/C_{0} \), we obtain the standard form of direct proportionality.

Analytically, this form is required to satisfy the property of additivity. If we assume that brewing two batches of tea bags, \( x \) and \( y \), separately or together should require the same total volume of water, the function must satisfy the Cauchy functional equation:

$$f(x + y) = f(x) + f(y)$$

Mathematically, for functions that are continuous or bounded on an interval, as is expected for physical volumes, the only solution is \( f(x) = kx \). This derivation ensures the infusion’s strength remains invariant regardless of the quantity prepared. However, while this model satisfies additivity, it must still be evaluated against the requirement of homogeneity to be considered a true linear transformation.

The Challenge of Homogeneity #

A linear transformation must satisfy two fundamental axioms: additivity and homogeneity.

1. Additivity: \( f(x + y) = f(x) + f(y) \)
2. Homogeneity: \( f(cx) = c f(x) \) for any scalar \( c \).

While additivity holds for quantities of tea, homogeneity presents a formal hurdle. In linear algebra, the scalar \( c \) must come from a field, such as the real numbers \( \mathbb{R} \), which includes negative values.

If we attempt to apply a negative scalar to our tea function, such as \( c = -2 \), the expression \( f(-2x) \) becomes physically meaningless. We cannot have a negative number of tea bags or a negative volume of water. Because the domain \( \mathbb{R}_{\ge 0} \) is not closed under multiplication by negative scalars, the function fails the strict test for a linear transformation.

Functions on Convex Cones #

Since the tea bag model doesn’t fit the requirements of a vector space, we can categorize it as a function on a convex cone. A convex cone is a subset of a vector space that is closed under addition and multiplication by positive scalars only.

Functions that are additive and positively homogeneous are prevalent in daily life:

• Recipe Scaling: If a specific recipe for bread requires 500g of flour per loaf, the total flour needed scales linearly with the number of loaves.
• Fuel Consumption: In a vehicle maintaining constant efficiency, the fuel consumed is proportional to the distance traveled. Traveling zero kilometers consumes zero liters, and every additional kilometer adds a fixed amount of fuel.
• Unit Pricing in Commerce: When purchasing identical items at a fixed price—such as five identical bolts at a hardware store—the total cost is additive. Buying two bolts and then three bolts costs the same as buying five at once.

In conclusion, while your tea preparation is linear in everyday terms, it is formally a positive linear map on a convex cone. It respects the logic of scaling and addition, but it stops where the negative domain of abstract vector spaces begins.