Problem 14 - Differential Equations and Linear Algebra

Question

Determine a formula for the linear transformation meeting the given conditions. $$\begin{aligned} &T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2} \text { such that }\\\ &T\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=(2,-5), T\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=(0,-3)\\\ &T\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]=(1,1), T\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right]=(-6,2) \end{aligned}$$

Step-by-Step Solution

Answer

The formula for the given linear transformation $$T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}$$, based on the given conditions, is: $$T(A) = [a(1 - 2db) + 2b, -3c + 2d + b - 5db]$$, where $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_{2}(\mathbb{R})$$.

Step 1: Create the transformation formula

Step 1: Recall that linearity means that the following two properties hold for the transformation $$T$$: 1. $$T(cA) = c(TA)$$ for any $$A \in M_{2}(\mathbb{R})$$ and any scalar $$c \in \mathbb{R}$$. 2. $$T(A + B) = T(A) + T(B)$$ for any $$A, B \in M_{2}(\mathbb{R})$$. Step 2: Notice that we can represent any matrix in $$M_{2}(\mathbb{R})$$ as a linear combination of the given matrices: $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + d\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} + b\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} - db\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ Step 3: Use the linearity properties to find the effect of the transformation on any matrix $$A \in M_{2}(\mathbb{R})$$: $$T(A) = T\left(a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + d\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} + b\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} - db\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\right)$$ Apply the linearity properties to compute $$T(A)$$: $$T(A) = aT\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + cT\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + dT\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} + bT\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} - dbT\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ Step 4: Substitute the given conditions into the expression for $$T(A)$$: $$T(A) = a(1, 1) + c(0, -3) + d(-6, 2) + b(2, -5) - db(2, -5)$$ Rearrange the terms to combine the coefficients: $$T(A) = [ad + a + 2b - 2db, -3c + 2d + b - 5db] = [a(1 - 2db) + 2b, -3c + 2d + b - 5db]$$ Step 5: We have found the formula for the given linear transformation $$T$$: $$T(A) = [a(1 - 2db) + 2b, -3c + 2d + b - 5db]$$

Step 2: Continue the solution

The formula for the given linear transformation $$T: M_{2}(\mathbb{R}) \rightarrow \mathbb{R}^{2}$$, based on the given conditions, is:

Step 3: Continue the solution

$$T(A) = [a(1 - 2db) + 2b, -3c + 2d + b - 5db]$$, where $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_{2}(\mathbb{R})$$.

Key Concepts

Matrix Algebra

Matrix Algebra serves as a fundamental component in understanding linear transformations. At its core, matrix algebra involves the manipulation and combination of matrices, using operations such as addition, subtraction, and multiplication. These operations are essential when forming linear combinations of matrices. In the context of linear transformations, such as those seen in the provided exercise, we deal with these operations to represent transformations from one space to another.
Matrix algebra allows us to express complex transformations compactly by using a matrix representation, which simplifies the application of linearity rules. However, it's crucial to handle the matrix operations with precision to ensure the correctness of the transformation formula.
Understanding matrix algebra is pivotal as it sets the stage for grasping more advanced topics in linear transformations, like linearity properties and the application of transformation formulas.

Linearity Properties

Linearity Properties are a vital concept for understanding linear transformations. A linear transformation must satisfy two basic properties: scalar multiplication and additivity.

• Scalar Multiplication Property: This property states that scaling a matrix by a factor 'c' should also scale the transformation result by 'c'. Mathematically, if the transformation is represented as \(T\) and the matrix as \(A\), then \(T(cA) = cT(A)\).
• Additivity Property: This property dictates that the transformation of a sum of matrices should be the sum of their individual transformations, i.e., \(T(A + B) = T(A) + T(B)\), where \(A\) and \(B\) are matrices.

These properties ensure that the transformation behaves predictably under matrix operations. They are pivotal for constructing and understanding the transformation formula, as they allow us to break down and manipulate the transformation of any arbitrary matrix using the transformations of predefined basis matrices.

Transformation Formula

The Transformation Formula is the core result that characterizes how each matrix in the preimage space is mapped to a vector in the image space. In the given exercise, you needed to find such a formula for a particular linear transformation \(T\) from \(M_{2}(\mathbb{R})\) to \(\mathbb{R}^2\).
To derive this formula, matrix algebra and the linearity properties come into play. By expressing each matrix as a linear combination of basis matrices, one can apply the linearity properties to determine how each component is transformed. For example, the given transformation conditions allowed for direct substitution and manipulation to arrive at a formula like:
\[T(A) = [a(1 - 2db) + 2b, -3c + 2d + b - 5db]\]
This formula shows how any matrix \(A\) is linearly transformed into a vector in \(\mathbb{R}^2\), indicating a specific pattern or rule derived from the transformation characteristics given in the problem.

Matrix Representation

Matrix Representation is a method that allows us to embody a linear transformation as a single matrix operation. In essence, it is the matrix form of a transformation, offering a compact and efficient way to calculate the result of transformations.
In the given exercise, the transformation \(T\) takes matrices from \(M_{2}(\mathbb{R})\) to vectors in \(\mathbb{R}^2\). Finding a matrix representation involves determining how basic matrices are individually transformed and combining these transformations into a single matrix.
This representation is powerful: once you have it, you can easily compute the transformation of any matrix within the domain by simple matrix multiplication. It simplifies complex procedures into straightforward computations and is essential for many applications in linear algebra. The goal of representing transformations with matrices is to enable various mathematical and computational analyses with higher efficiency and clarity.