Problem 15 - Differential Equations and Linear Algebra

Question

Determine the matrix of the given transformation $$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$$. $$T\left(x_{1}, x_{2}\right)=\left(x_{1}+3 x_{2}, 2 x_{1}-7 x_{2}, x_{1}\right)$$.

Step-by-Step Solution

Answer

The matrix representation of the given linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) is: \[A = \begin{bmatrix} 1 & 3 \\ 2 & -7 \\ 1 & 0 \end{bmatrix}\]

Step 1: 1. Identify the basis of domain and codomain

The domain and codomain of the transformation are \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), respectively. The standard basis for \(\mathbb{R}^{2}\) is \(\{\mathbf{e_{1}}, \mathbf{e_{2}}\}\), where \(\mathbf{e_{1}}=\begin{bmatrix}1 \\ 0 \end{bmatrix}\) and \(\mathbf{e_{2}}=\begin{bmatrix}0 \\ 1 \end{bmatrix}\).

Step 2: 2. Apply the transformation to each basis vector

Apply the transformation \(T\) to each basis vector \(\mathbf{e_{1}}\) and \(\mathbf{e_{2}}\): \(T(\mathbf{e_{1}}) = T(1, 0) = (1 + 3 \times 0, 2 - 7 \times 0, 1) = (1, 2, 1)\) \(T(\mathbf{e_{2}}) = T(0, 1) = (0 + 3 \times 1, 0 - 7 \times 1, 0) = (3, -7, 0)\)

Step 3: 3. Form the matrix of the transformation

To form the matrix of the transformation, we put the transformed basis vectors as columns: The matrix representation of the transformation \(T\) is: \[A = \begin{bmatrix} 1 & 3 \\ 2 & -7 \\ 1 & 0 \end{bmatrix}\] This matrix represents the given linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\).

Key Concepts

Basis Vectors

In linear algebra, basis vectors are fundamental building blocks in vector spaces. To understand them, consider them as a set of vectors that are used to express any vector in that space uniquely. In our exercise, we work with the standard basis vectors for \(\mathbb{R}^{2}\), which are:

• \(\mathbf{e_{1}} = \begin{bmatrix} 1 \ 0 \end{bmatrix}\)
• \(\mathbf{e_{2}} = \begin{bmatrix} 0 \ 1 \end{bmatrix}\)

Using these basis vectors, any vector in \(\mathbb{R}^{2}\) can be written as a linear combination of \(\mathbf{e_{1}}\) and \(\mathbf{e_{2}}\). This means that if you have a vector \(\mathbf{v} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\), it can be expressed as \(x_1 \cdot \mathbf{e_{1}} + x_2 \cdot \mathbf{e_{2}}\). These basis vectors form the simplest way to describe the space and are crucial for applying linear transformations.

Matrix Representation

The matrix representation of a linear transformation is the key to understanding how it operates on vectors. In essence, the matrix translates a transformation into a form that can be easily manipulated using matrix operations. For the transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) described in the exercise, we first apply the transformation to each basis vector of \(\mathbb{R}^{2}\).

The transformation of \(\mathbf{e_{1}}\) results in the vector \(T(\mathbf{e_{1}}) = (1, 2, 1)\) and the transformation of \(\mathbf{e_{2}}\) gives \(T(\mathbf{e_{2}}) = (3, -7, 0)\). To form the matrix representation, we align these vectors as columns into a matrix thus creating:

\[A = \begin{bmatrix} 1 & 3 \ 2 & -7 \ 1 & 0 \end{bmatrix}\]

This matrix \(A\) effectively encodes the transformation and allows us to transform any vector from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{3}\) through matrix multiplication.

Linear Transformation

A linear transformation is a function between two vector spaces that preserves addition and scalar multiplication. This means that if you take two vectors, say \(\mathbf{u}\) and \(\mathbf{v}\), and a scalar \(c\), the transformation \(T\) will satisfy:

• \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\)
• \(T(c \cdot \mathbf{u}) = c \cdot T(\mathbf{u})\)

The transformation given in the exercise, \(T(x_1, x_2) = (x_1 + 3x_2, 2x_1 - 7x_2, x_1)\), is linear because it adheres to these properties. Such transformations are significant because they can be completely understood and solved using matrix forms. This characteristic makes linear transformations a powerful tool in various mathematical and real-world applications, such as computer graphics, physics simulations, and data transformations, by simplifying complex operations into manageable computations.