Problem 5 - Differential Equations and Linear Algebra

Question

Verify directly from Definition 6.1 .3 that the given mapping is a linear transformation. \(T: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\) defined by $$ T(A)=A B-B A $$ where \(B\) is a fixed \(n \times n\) matrix.

Step-by-Step Solution

Answer

The given mapping \(T: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\), defined by \(T(A) = AB - BA\), where \(B\) is a fixed \(n \times n\) matrix, is a linear transformation since it satisfies both the additive and homogeneity properties. We showed this by verifying that \(T(A + C) = T(A) + T(C)\) and \(T(kA) = kT(A)\) for any matrices \(A, C \in M_{n}(\mathbb{R})\) and any scalar \(k\).

Step 1: Verify the Additive Property

We need to show that for any matrices \(A, C \in M_{n}(\mathbb{R})\): \(T(A + C) = T(A) + T(C)\). Let's compute \(T(A + C)\): \begin{align*} T(A + C) &= (A + C)B - B(A + C) \\ &= AB + CB - BA - BC \\ &= (AB - BA) + (CB - BC) \\ &= T(A) + T(C) \end{align*} Thus, the additive property holds true.

Step 2: Verify the Homogeneity Property

We need to show that for any scalar \(k\) and matrix \(A \in M_{n}(\mathbb{R})\): \(T(kA) = kT(A)\). Let's compute \(T(kA)\): \begin{align*} T(kA) &= (kA)B - B(kA) \\ &= k(AB) - k(BA) \\ &= k(AB - BA) \\ &= kT(A) \end{align*} Thus, the homogeneity property holds true. Since both the additive and homogeneity properties are satisfied, we can conclude that the given mapping \(T: M_{n}(\mathbb{R}) \rightarrow M_{n}(\mathbb{R})\), defined by \(T(A) = AB - BA\), where \(B\) is a fixed \(n \times n\) matrix, is indeed a linear transformation.

Key Concepts

Understanding the Additive Property

The additive property is a fundamental characteristic of linear transformations. It states that the transformation of a sum of inputs is equal to the sum of their individual transformations. In simpler terms, if you have two matrices, say \(A\) and \(C\), and you apply the transformation to their sum \(A+C\), the result should be identical to individually transforming \(A\) and transforming \(C\) and then adding those results.

Here's how it works for the given mapping: Let's calculate for matrices \(A\) and \(C\) in \(M_n(\mathbb{R})\):

• We start by adding \(A\) and \(C\) to get \(A+C\).
• Apply the transformation which is computed as \((A+C)B - B(A+C)\).
• After simplifying, we end up proving that \(T(A+C) = T(A) + T(C)\).

Breaking it down explicitly verifies that the additive property holds for this transformation, ensuring it behaves as a linear transformation.

Exploring the Homogeneity Property

The homogeneity property is another crucial property for linear transformations, often called scalar multiplication property. It asserts that scaling the input of a linear transformation scales the output by the same factor. In other words, if you multiply a matrix \(A\) by a scalar \(k\) and then apply the transformation, it's the same as applying the transformation first and then scaling the result by \(k\).

To see this in action with the transformation \(T(A) = AB - BA\):

• Take a matrix \(A\) and multiply it by a scalar \(k\), resulting in \(kA\).
• Apply the transformation to \(kA\), giving \((kA)B - B(kA)\).
• Simplify to get \(k(AB) - k(BA) = k(AB - BA)\), which is the same as \(kT(A)\).

Therefore, this direct calculation confirms the homogeneity property for our transformation, reinforcing its linear nature.

Diving into Matrix Algebra

Matrix algebra is the mathematical framework that deals with matrices, their properties, and the operations you can perform on them. In our context, it provides the tools to verify properties like the additive and homogeneity properties of transformations.

Several operations are fundamental in matrix algebra:

• Addition: Combining matrices by adding corresponding elements.
• Multiplication: Involves computing the dot product of rows and columns, resulting in a new matrix. It's important to remember that matrix multiplication is not commutative, i.e., \(AB eq BA\) in general.
• Scalars: Multiplying every entry of a matrix by a scalar value, scaling the entire matrix.

These operations follow specific rules and properties, some of which are parallel to familiar arithmetic rules, while others differ, like the non-commutativity of matrix multiplication. Understanding these principles is crucial to grasping how linear transformations work. Each step in verifying the transformation \(T(A) = AB - BA\) leverages these matrix algebra concepts to ensure the transformation is, indeed, linear.