Question

Show that a linear combination of two derivations is a derivation.

Short answer

The linear combination \(D = aD_1 + bD_2\) of two derivations \(D_1\) and \(D_2\) is a derivation, as it is a linear map and satisfies Leibniz's law.

Step-by-step solution

Represent the linear combination of two derivations

Assume we have two derivations, \(D_1\) and \(D_2\), and scalar constants \(a\) and \(b\). The linear combination of these two derivations can be represented as \(D = aD_1 + bD_2\).

Show that our linear combination is a linear map

To do this, take arbitrary vectors \(v\) and \(w\), and scalar \(c\). We need to show that \(D(cv + w) = cDv + Dw\). Since \(D_1\) and \(D_2\) are derivations, which means they are linear maps, we know that \(D(cv + w) = aD_1(cv + w) + bD_2(cv + w) = acD_1v + aD_1w + bcD_2v + bD_2w = cDv + Dw\). Therefore, the linear combination is a linear map.

Show that our linear combination satisfies Leibniz's law

Similarly, for two arbitrary vectors \(u\) and \(v\), we need to show that \(D(uv) = u(Dv) + v(Du)\). We again use the fact that \(D_1\) and \(D_2\) are derivations. So, \(D(uv) = aD_1(uv) + bD_2(uv) = u(aD_1v + bD_2v) + v(aD_1u + bD_2u) = uDv + vDu\). Hence the linear combination also satisfies Leibniz's law.

Key concepts

Understanding Linear Combination in Mathematics

A linear combination in algebra is a fundamental concept which involves the combination of several terms, typically vectors or functions, multiplied by scalar coefficients and added together. The equation can be represented as follows: \begin{align*}v = a_1v_1 + a_2v_2 + \ ... + a_nv_n,\text{where } a_i \text{ are scalars, and } v_i \text{ are vectors or functions.}\text{In the context of the exercise, we consider derivations.}\text{Derivations are special functions that capture the idea of a derivative in certain mathematical settings, such as in abstract algebra. A derivation is linear if it satisfies the following property for any scalars } c \text{ and vectors } v,w:\[D(cv + w) = cD(v) + D(w).\]\text{When we combine two derivations, say } D_1 \text{ and } D_2, \text{ with constants } a \text{ and } b \text{ respectively, we get a new function } D \text{ that is their linear combination:}\[D = aD_1 + bD_2.\]\text{The goal is to show that this new function } D \text{ is also a derivation, which means it must keep up the properties of linearity and satisfy Leibniz's law, just as the original derivations } D_1 \text{ and } D_2 \text{ do.}

Exploring Linear Maps and their Role in Mathematics

Linear maps, also known as linear transformations or linear functions, are critical in understanding not just algebra, but multiple areas of mathematics. A map 'L' is called linear if it satisfies two main properties for all vectors 'v' and 'w', and any scalar 'c':

• Additivity: \(L(v + w) = L(v) + L(w)\)
• Homogeneity: \(L(cv) = cL(v)\)

Throughout the exercise improvement process, it's important to clearly illustrate that a linear combination of linear maps retains these properties. As seen in the step-by-step solution, to prove our function 'D' is a linear map, you take arbitrary vectors 'v' and 'w' and a scalar 'c', and verify the criteria above. If 'D' satisfies these conditions as a result of being a linear combination of derivations—which inherently are linear maps—'D' thereby upholds the hallmark properties of linearity. Emphasizing examples and showcasing the simple mechanical steps to verify linearity can concretize this concept for students.

Deciphering Leibniz's Law within Derivations

Leibniz's law, named after the mathematician Gottfried Wilhelm Leibniz, plays a crucial role particularly in the realm of differential calculus and its applications in various algebraic structures. It is essentially the property that describes how a derivation behaves with respect to the product of two functions or elements. In its classic form for derivatives, it can be stated as:\[D(uv) = uD(v) + vD(u),\]where 'D' is the derivation, and 'u' and 'v' are differentiable functions or elements in a specific algebraic structure. In the scope of our exercise, we need to ensure that the linear combination of derivations adheres to this law. The step-by-step solution shows that 'D', the linear combination, does satisfy Leibniz's law when applied to the product of arbitrary elements 'u' and 'v'.
For students, breaking down this concept with concrete examples where 'u' and 'v' are chosen distinctly can help in understanding how the product rule is applied in the context of derivations, and why it is vital for 'D' to be a derivation itself.