Question

Show that if \(\mathbf{F}\) is continuous on \(\mathbb{R}^{n}\) and \(F(\mathbf{X}+\mathbf{Y})=\mathbf{F}(\mathbf{X})+\mathbf{F}(\mathbf{Y})\) for all \(\mathbf{X}\) and \(\mathbf{Y}\) in \(\mathbb{R}^{n},\) then \(\mathbf{A}\) is linear. HINT: The rational numbers are dense in the reals.

Short answer

Question: Show that a continuous function $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ satisfying $F(\mathbf{X} + \mathbf{Y}) = \mathbf{F}(\mathbf{X}) + \mathbf{F}(\mathbf{Y})$ for all $\mathbf{X}$ and $\mathbf{Y} \in \mathbb{R}^n$ is a linear transformation. Solution: We proved that $F$ satisfies the two properties for linearity: $F(c\mathbf{X}) = cF( \mathbf{X})$ for all $c \in \mathbb{R}$ and $\mathbf{X} \in \mathbb{R}^n$ and $F(\mathbf{X} + \mathbf{Y}) = F(\mathbf{X}) + F(\mathbf{Y})$ for all $\mathbf{X}, \mathbf{Y} \in \mathbb{R}^n$. Therefore, $F$ is a linear transformation.

Step-by-step solution

Prove the linearity property for rational multiples

We first show that the linearity property holds for rational multiples, i.e., \(\mathbf{F}(q\mathbf{X}) = q\mathbf{F}(\mathbf{X})\) for all \(q \in \mathbb{Q}\) and \(\mathbf{X} \in \mathbb{R}^n\). To do this, we can use induction on the summation property of \(\mathbf{F}\): - For \(q = 1\), the property clearly holds: \(\mathbf{F}(1\mathbf{X}) = \mathbf{F}(\mathbf{X})\). - Suppose the property holds for some \(q=k\): $$\mathbf{F}(k\mathbf{X}) = k\mathbf{F}(\mathbf{X})$$ - Now we show that it holds for \(q=k+1\): \begin{align*} \mathbf{F}((k+1)\mathbf{X}) &= \mathbf{F}(k\mathbf{X} + \mathbf{X}) \\ &= \mathbf{F}(k\mathbf{X}) + \mathbf{F}(\mathbf{X}) \\ &= k\mathbf{F}(\mathbf{X}) + \mathbf{F}(\mathbf{X}) \\ &= (k+1)\mathbf{F}(\mathbf{X}) \end{align*} Thus, by the principle of induction, the property holds for all rational multiples.

Extend the linearity property to real multiples

Now we will show that the linearity property holds for real multiples as well. Pick some \(c \in \mathbb{R}\) and \(\mathbf{X} \in \mathbb{R}^n\). We are given that \(\mathbf{F}\) is continuous on \(\mathbb{R}^n\). Since rational numbers are dense in the reals, there exists a sequence of rational numbers \((q_n)\) such that \(q_n \to c\) as \(n \to \infty\). By the continuity of \(\mathbf{F}\), we have $$\lim_{n \to \infty} \mathbf{F}(q_n\mathbf{X}) = \mathbf{F}(c\mathbf{X})$$ From Step 1, we know that \(\mathbf{F}(q_n\mathbf{X}) = q_n\mathbf{F}(\mathbf{X})\). So, we can now rewrite the limit as $$\lim_{n \to \infty} q_n\mathbf{F}(\mathbf{X}) = \mathbf{F}(c\mathbf{X})$$ The left side is equal to \(c\mathbf{F}(\mathbf{X})\) since \(q_n \to c\). Therefore, we have shown that \(\mathbf{F}(c\mathbf{X}) = c\mathbf{F}(\mathbf{X})\) for all \(c \in \mathbb{R}\) and \(\mathbf{X} \in \mathbb{R}^n\).

Conclusion

We have established the two properties required for \(\mathbf{F}\) to be a linear transformation: 1. \(\mathbf{F}(c\mathbf{X}) = c\mathbf{F}(\mathbf{X})\) for all \(c \in \mathbb{R}\) and \(\mathbf{X} \in \mathbb{R}^n\) 2. \(\mathbf{F}(\mathbf{X} + \mathbf{Y}) = \mathbf{F}(\mathbf{X}) + \mathbf{F}(\mathbf{Y})\) for all \(\mathbf{X}, \mathbf{Y} \in \mathbb{R}^n\) (given) Thus, we can conclude that \(\mathbf{F}\) is indeed a linear transformation.

Key concepts

Continuity

Continuity is a fundamental concept in mathematical analysis, especially in the study of functions. In simple terms, a function is said to be continuous if small changes in its input result in small changes in its output. Mathematically, a function \( f \) is continuous at a point \( x_0 \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - x_0| < \delta \), it follows that \( |f(x) - f(x_0)| < \epsilon \). This definition captures the idea that the graph of a continuous function has no jumps, breaks, or holes.

In the context of the exercise, the continuity of the function \( \mathbf{F} \) is crucial. It allows the linearization of \( \mathbf{F} \) from rational multipliers to real multipliers due to the density of rational numbers in the real numbers. Because \( \mathbf{F} \) is continuous, any limit of the rational approximation sequence is guaranteed to match the value of \( \mathbf{F} \) at the real point. This property ensures that the adjustments in input reflect proportional changes in output smoothly over both rational and real numbers.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where \( p \) is the numerator, \( q \) is the denominator, and \( q eq 0 \). Examples include \( \frac{1}{2} \), \( 0.75 \), and \(-3\). Rational numbers form a dense subset of the real numbers, which means that between any two real numbers, there is a rational number. This property is significant in analysis because it allows us to approximate real numbers using rational numbers.

The exercise leverages this density property. Since the linearity of \( \mathbf{F} \) has been shown for rational multiples, and rational numbers are dense, we can approximate any real number with a sequence of rational numbers. As these rational numbers approach a real number, their corresponding function values exhibit a similar convergence because of \( \mathbf{F} \)'s continuity. This step effectively bridges the gap between rational multiple inputs of the function and real multiple inputs, ensuring the function's linearity holds over all real numbers.

Real Analysis

Real analysis is a branch of mathematics dealing with real numbers and real-valued functions. It investigates properties like limits, continuity, derivatives, and integrals. In real analysis, understanding the behavior of functions and sequences is essential. It provides tools and frameworks, such as epsilon-delta definitions, to rigorously prove the properties of functions.

The exercise in question demonstrates a classic real analysis problem where we transition from demonstrating a property with rational numbers to extending it to all real numbers. The function \( \mathbf{F} \) is shown to have linearity properties when its inputs are rationals, and by using the construction from real analysis—especially continuity and the density of rational numbers—we extend these properties to real numbers.

This exemplifies foundational real analysis techniques, where proving results often involves using sequences and limits. By considering how \( \mathbf{F} \) behaves under a sequence of rational approximations, we apply a real analysis approach to ensure the argument holds universally for all real numbers, illustrating the power and elegance of real analysis in bridging abstract mathematical concepts.