Problem 19 - Differential Equations and Linear Algebra

Question

Determine the linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) that has the given matrix. $$A=\left[\begin{array}{rr} 1 & 3 \\ -4 & 7 \end{array}\right]$$.

Step-by-Step Solution

Answer

The linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) with the given matrix \(A = \left[\begin{array}{rr} 1 & 3 \\\ -4 & 7 \end{array}\right]\) can be determined as \(T(x) = Ax = \begin{bmatrix}1x_1+3x_2 \\ -4x_1+7x_2\end{bmatrix}\). So, \(T:\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \mapsto \begin{bmatrix}1x_1+3x_2 \\ -4x_1+7x_2\end{bmatrix}\).

Step 1: Find the dimensions of the domain and codomain

We are given the matrix A, which dictates how the linear transformation T operates on the vectors from the domain: $$A=\left[\begin{array}{rr} 1 & 3 \\\ -4 & 7 \end{array}\right]$$ First, let's identify the dimensions of the domain (\(n\)) and codomain (\(m\)). Observe that A is a 2x2 square matrix, so the linear transformation T: \(\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) is given by \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\).

Step 2: Determine the linear transformation

Now, let's determine the linear transformation T. We know that A acts on input vectors and transforms them to a new output. In other words, if we have an input vector \(x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}\), then the output vector, also known as the image of x under T, can be written as \(T(x) = Ax\). Explicitly, we compute \(T(x) = Ax\) by multiplying matrix A by vector x: $$T(x) = Ax = \left[\begin{array}{rr} 1 & 3 \\\ -4 & 7 \end{array}\right]\begin{bmatrix} x_1\\ x_2\end{bmatrix}$$

Step 3: Perform the matrix-vector multiplication

Multiply matrix A by the vector x and simplify: $$T(x) = \begin{bmatrix} 1x_1+3x_2 \\ -4x_1+7x_2\end{bmatrix}$$ The result indicates that the linear transformation T is given by: $$T:\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \mapsto \begin{bmatrix}1x_1+3x_2 \\ -4x_1+7x_2\end{bmatrix}$$ This linear transformation T: \(\mathbb{R}^2 \rightarrow \mathbb{R}^2\) has the given matrix A and maps input vectors in \(\mathbb{R}^2\) to output vectors in \(\mathbb{R}^2\).

Key Concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a fundamental operation in linear algebra. It allows us to apply linear transformations to vectors, which can be thought of as moving points in space or changing their properties like direction or length. To perform matrix-vector multiplication, you'll need a matrix and a vector.
The matrix is usually represented as a 2D array of numbers, and the vector is a single column or row of numbers. In the context of our exercise, the matrix is matrix \(A\), with dimensions \(2\times2\), and is written as:
\[A=\begin{bmatrix} 1 & 3 \ -4 & 7 \end{bmatrix}\]The vector, on the other hand, is denoted as \(x\), with entries \(x_1\) and \(x_2\), and is a column vector:
\[x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]The process involves taking each element of the vector and multiplying it with its corresponding row in the matrix. Then, you sum the results to form the new entries of the resulting vector. This operation can be represented as:

• The first entry of the new vector is calculated as \(1x_1 + 3x_2\).
• The second entry is \(-4x_1 + 7x_2\).

Thus, the product \(Ax\) yields the new vector:
\[T(x) = \begin{bmatrix} 1x_1+3x_2 \ -4x_1+7x_2 \end{bmatrix}\] Matrix-vector multiplication is crucial because it harnesses the power of matrices to encode and apply transformations to vectors, allowing us to understand and manipulate data effectively in both theoretical and applied contexts.

Matrix Representation

Matrix representation is a way to express linear transformations in terms of matrices. It provides a framework to simplify complex transformations into manageable operations involving additions and multiplications. In mathematics, this representation is powerful because it allows the application of algebraic techniques to solve complex problems.
In our exercise, the matrix \(A\) represents the linear transformation \(T\). The rows of the matrix correspond to different transformation rules applied to the input vector. For example, consider our matrix:
\[A = \begin{bmatrix} 1 & 3 \ -4 & 7 \end{bmatrix}\]

• The first row \([1, 3]\) indicates how the linear transformation modifies the first component of a vector.
• The second row \([-4, 7]\) conveys similar information for the second component.

Matrices like these encode how each part of the input vector contributes to the transformed output. This provides a concise and uniform way to write and compute transformations, thus making calculations easier, particularly with larger sets of data.
To understand and use matrix representation effectively, one should practice with various problems that involve constructing matrices from given transformations, as well as decoding what specific matrices imply about transformation properties, such as rotations, reflections, or scalings.

Domain and Codomain

The concepts of domain and codomain are vital in understanding linear transformations. The domain of a transformation is the set of all possible input values, while the codomain is the set of potential output values.
For the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) described in our exercise:

• The domain of \(T\) is \(\mathbb{R}^2\). This means that the transformation can take any vector from the 2-dimensional real space.
• The codomain is also \(\mathbb{R}^2\), implying that any output of the transformation will also lie within this space.

Understanding the domain and codomain is crucial because they define the 'shape' and 'reach' of a transformation. When you know these spaces, you can predict the behavior and limitations of the transformation better. For example, a transformation from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) would imply reduction in dimension, potentially "flattening" inputs.
Analyzing the dimensions of the domain and codomain through the transformation's matrix is essential, as it directly impacts real-world applications like computer graphics, where transformation maps 3D objects to 2D screens.