Problem 14 - Differential Equations and Linear Algebra

Question

For Problems \(14-18\), determine the matrix of the given transformation $$ T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m} $$ \(T\left(x_{1}, x_{2}\right)=\left(3 x_{1}-2 x_{2}, x_{1}+5 x_{2}\right)\)

Step-by-Step Solution

Answer

The matrix of the given transformation T from ℝ² to ℝ² is \(A = \begin{pmatrix} 3 & -2\\ 1 & 5 \end{pmatrix}\).

Step 1: Identify the input basis vectors e₁ and e₂

Since T is a linear transformation from ℝ^2 to ℝ^2, the input basis vectors are the standard basis vectors for ℝ^2, e₁ and e₂, given by: $$e_{1} = \begin{pmatrix} 1\\ 0 \end{pmatrix}, e_{2} = \begin{pmatrix} 0\\ 1 \end{pmatrix}$$

Step 2: Apply the transformation T to each basis vector

Now, we will apply the transformation T to both basis vectors: For e₁: $$T(e_{1}) = T(1, 0) = (3(1) - 2(0), 1(1) + 5(0)) = (3, 1)$$ For e₂: $$T(e_{2}) = T(0, 1) = (3(0) - 2(1), 1(0) + 5(1)) = (-2, 5)$$

Step 3: Construct the matrix of the given transformation T

Using the results of step 2, we can construct a matrix A of the linear transformation such that: $$ A = \begin{pmatrix} T(e_{1}) & T(e_{2}) \end{pmatrix} $$ Replacing the transformed vectors, we get: $$A = \begin{pmatrix} 3 & -2\\ 1 & 5 \end{pmatrix}$$ So, the matrix of the given transformation T is: $$A = \begin{pmatrix} 3 & -2\\ 1 & 5 \end{pmatrix}$$

Key Concepts

Matrix Representation

A matrix representation is a way of representing a linear transformation using a matrix. It's like a blueprint that tells us how a transformation works on vectors. When we have a transformation, such as a function that maps vectors from one space to another, we can represent it with a matrix so that calculations become more manageable. This representation is formed by observing how the transformation acts on the standard basis vectors of the space. The results of these transformations give the columns of the matrix. For example, if we know how our transformation affects each basis vector of a two-dimensional space, we can arrange these affected vectors as columns in a matrix. This matrix will then perform the same transformation when multiplied by other vectors in that space.

Standard Basis Vectors

Standard basis vectors are the building blocks of vector spaces. They are the simplest vectors possible, typically consisting of zeros in most positions and a single one in one position. For example, in two-dimensional space, the standard basis vectors are \( e_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix} \) and \( e_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix} \). These vectors form a base set from which any other vector in the space can be constructed through linear combinations. They are essential because they allow us to define and better understand transformations by simplifying the process. In linear algebra, transformations are often described by how they change these basis vectors, making it straightforward to construct a transformation matrix.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but most importantly, it studies linear transformations and how they can be represented by matrices. An essential part of linear algebra is understanding how spaces are related through transformations, which are expressed using matrices. This discipline provides the foundations for many fields, like engineering, physics, and computer science, thanks to its powerful tools for modeling and solving problems involving multiple variables. Key components in linear algebra include:

• Vectors and vector spaces
• Matrix operations and determinants
• Eigenvalues and eigenvectors
• Systems of linear equations

Transformation Matrix

A transformation matrix is a specific matrix that describes how a linear transformation acts on a vector space. It's like a recipe: given a vector to transform, the matrix describes the operations that will be applied. Transformation matrices are crucial in linear algebra because they provide a simple way to compute the effect of a transformation across an entire space. To find a transformation matrix, we apply the transformation to each standard basis vector and compile the results as columns of the matrix. For example, if a transformation \( T \) maps \( e_1 \) to \( (3, 1) \) and \( e_2 \) to \( (-2, 5) \), then the transformation matrix \( A \) is \( \begin{pmatrix} 3 & -2 \ 1 & 5 \end{pmatrix} \). When we multiply this matrix by another vector from the space, the result is the transformed vector. Thus, transformation matrices are efficient tools to understand and execute transformations systematically.