How to Prove a Set of Functions Forms a Vector Space (Step-by-Step)

📚 What is a Vector Space?

A vector space is a set of objects, called vectors, that can be added together and multiplied by scalars (usually real numbers) while still remaining within the same set. More formally, it's a set that satisfies a list of axioms, ensuring these operations behave in a predictable manner.

📜 A Little History

The concept of a vector space emerged gradually throughout the 19th century. Mathematicians like Hermann Grassmann and Giuseppe Peano laid the groundwork by formalizing the algebraic properties of vectors. The modern definition, encompassing functions and other abstract objects, became more prominent in the 20th century with the rise of functional analysis.

🔑 The Vector Space Axioms

To prove that a set of functions $V$ is a vector space under the operations of function addition and scalar multiplication, you must demonstrate that all the following axioms hold:

• ➕ Closure under addition: For all functions $f, g \in V$, their sum $(f+g)$ must also be in $V$. In other words, adding two functions from the set results in another function within the same set.
• 🔢 Associativity of addition: For all functions $f, g, h \in V$, $(f+g)+h = f+(g+h)$. The order in which you add doesn't change the result.
• 🆔 Existence of an additive identity: There exists a zero function, denoted by $0$, such that for all functions $f \in V$, $f + 0 = f$. This is a function that, when added to any other function in the set, leaves the other function unchanged.
• ➖ Existence of additive inverses: For every function $f \in V$, there exists a function $-f \in V$ such that $f + (-f) = 0$. This is the function that, when added to $f$, yields the zero function.
• 🤝 Commutativity of addition: For all functions $f, g \in V$, $f+g = g+f$. The order in which you add the functions does not matter.
• ✖️ Closure under scalar multiplication: For all functions $f \in V$ and any scalar $c$, the product $cf$ must also be in $V$. Multiplying a function from the set by a scalar results in another function within the same set.
• ⚖️ Distributivity of scalar multiplication with respect to vector addition: For all functions $f, g \in V$ and any scalar $c$, $c(f+g) = cf + cg$.
• ➕ Distributivity of scalar multiplication with respect to scalar addition: For all functions $f \in V$ and any scalars $c, d$, $(c+d)f = cf + df$.
• 🎯 Associativity of scalar multiplication: For all functions $f \in V$ and any scalars $c, d$, $c(df) = (cd)f$.
• 1️⃣ Existence of a multiplicative identity: For all functions $f \in V$, $1f = f$. Multiplying by the scalar 1 leaves the function unchanged.

📝 Step-by-Step Proof: The Even Polynomials Example

Let's prove that the set $P_e$ of all even polynomials (polynomials where $f(-x) = f(x)$) forms a vector space.

1. Define the Set: $P_e = \{f(x) \mid f(x) = a_0 + a_2x^2 + a_4x^4 + ... + a_{2n}x^{2n}, a_i \in \mathbb{R}\}$
2. Closure under addition: Let $f(x), g(x) \in P_e$. Then $f(-x) = f(x)$ and $g(-x) = g(x)$. Now, $(f+g)(x) = f(x) + g(x)$, so $(f+g)(-x) = f(-x) + g(-x) = f(x) + g(x) = (f+g)(x)$. Therefore, $f+g \in P_e$.
3. Closure under scalar multiplication: Let $f(x) \in P_e$ and $c \in \mathbb{R}$. Then $f(-x) = f(x)$. Now, $(cf)(x) = c \cdot f(x)$, so $(cf)(-x) = c \cdot f(-x) = c \cdot f(x) = (cf)(x)$. Therefore, $cf \in P_e$.
4. Additive Identity: The zero function, $0(x) = 0$, is an even polynomial since $0(-x) = 0 = 0(x)$.
5. Additive Inverse: If $f(x) \in P_e$, then $-f(x)$ is also in $P_e$ because if $f(-x) = f(x)$, then $-f(-x) = -f(x)$.
6. Other Axioms: The remaining axioms (associativity, commutativity, distributivity, multiplicative identity) follow directly from the properties of polynomial addition and scalar multiplication.

Since all the axioms are satisfied, the set of even polynomials, $P_e$, forms a vector space.

🌍 Real-World Examples

• 🎵 Audio Signal Processing: The set of all audio signals of a fixed duration can be considered a vector space. Signal processing techniques often rely on linear combinations of signals, which are vector space operations.
• 📸 Image Processing: Images can be represented as matrices, and the set of all images of a fixed size can be treated as a vector space. Operations like image blending and filtering involve scalar multiplication and addition.
• 🌡️ Temperature Distributions: The set of all possible temperature distributions in a room can be modeled as a vector space. Combining temperature profiles or scaling them are valid vector space operations.

💡 Conclusion

Proving a set of functions forms a vector space involves systematically verifying each of the vector space axioms. By carefully checking these conditions, you can confidently determine whether a given set of functions possesses the necessary structure to be considered a vector space. This understanding is crucial in many areas of mathematics, physics, and engineering.