Problem 20 - Differential Equations and Linear Algebra

Question

Determine the linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) that has the given matrix. $$A=\left[\begin{array}{rrr} 2 & -1 & 5 \\ 3 & 1 & -2 \end{array}\right]$$.

Step-by-Step Solution

Answer

The linear transformation \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^2\) corresponding to the given matrix A is: \[T(\bold{x}) = \begin{bmatrix} 2x_1 - x_2 + 5x_3 \\ 3x_1 + x_2 - 2x_3 \end{bmatrix} \]

Step 1: Identify the standard basis vectors for the domain

The standard basis vectors for the domain \(\mathbb{R}^3\) are: \begin{align*} \bold{e}_1 = \begin{bmatrix} 1\\0\\0 \end{bmatrix}, \quad \bold{e}_2 = \begin{bmatrix} 0\\1\\0 \end{bmatrix}, \quad \bold{e}_3 = \begin{bmatrix} 0\\0\\1 \end{bmatrix} \end{align*}

Step 2: Apply the matrix A to each basis vector

We now find the image of each basis vector under the matrix A. In other words, we need to compute the product \(A\bold{e}_i\) for \(i=1,2,3\). \begin{align*} A\bold{e}_1 = \begin{bmatrix} 2 & -1 & 5 \\ 3 & 1 & -2 \end{bmatrix} \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 2\\3 \end{bmatrix} \end{align*} \begin{align*} A\bold{e}_2 = \begin{bmatrix} 2 & -1 & 5 \\ 3 & 1 & -2 \end{bmatrix} \begin{bmatrix} 0\\1\\0 \end{bmatrix} = \begin{bmatrix} -1\\1 \end{bmatrix} \end{align*} \begin{align*} A\bold{e}_3 = \begin{bmatrix} 2 & -1 & 5\\ 3 & 1 & -2 \end{bmatrix} \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 5\\-2 \end{bmatrix} \end{align*}

Step 3: Define the linear transformation T

Now we can define the linear transformation T using the results from the previous step: For any vector \(\bold{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3\), let: \[ T(\bold{x}) = T(x_1\bold{e}_1 + x_2\bold{e}_2 + x_3\bold{e}_3) = x_1 A\bold{e}_1 + x_2 A\bold{e}_2 + x_3 A\bold{e}_3 \] Substitute the images we computed earlier: \[ T(\bold{x}) = x_1 \begin{bmatrix} 2\\3 \end{bmatrix} + x_2 \begin{bmatrix} -1\\1 \end{bmatrix} + x_3 \begin{bmatrix} 5\\-2 \end{bmatrix} \] The linear transformation T corresponding to the given matrix A is: \[ T(\bold{x}) = \begin{bmatrix} 2x_1 - x_2 + 5x_3 \\ 3x_1 + x_2 - 2x_3 \end{bmatrix} \]

Key Concepts

Matrix Representation

Matrix representation is an essential aspect of understanding linear transformations. A matrix, like the one given in this exercise, functions as a compact way to represent such transformations between two vector spaces. Specifically, it transforms vectors from one space, say \( \mathbb{R}^3 \), to another, \( \mathbb{R}^2 \). In our example, we have a matrix \( A \) of size 2x3, which indicates that it takes a vector from a three-dimensional space and projects it onto a two-dimensional space.

• Each column of the matrix corresponds to the image of one of the standard basis vectors under the transformation.
• The number of rows shows how many dimensions the image has.
• Matrix-vector multiplication is employed to apply this transformation to any vector in the domain.

This powerful concept allows us to simplify and work efficiently with complex transformations in linear algebra.

Standard Basis Vectors

Standard basis vectors are the building blocks of vector spaces. In our case, the standard basis vectors for \(\mathbb{R}^3\) are:

• \( \bold{e}_1 = \begin{bmatrix} 1\0\0 \end{bmatrix} \)
• \( \bold{e}_2 = \begin{bmatrix} 0\1\0 \end{bmatrix} \)
• \( \bold{e}_3 = \begin{bmatrix} 0\0\1 \end{bmatrix} \)

These vectors serve multiple purposes in the context of linear transformations:

• They help in understanding how the transformation affects the entire space by examining its effect on each of these simple vectors.
• By applying the transformation matrix \( A \) to these vectors, we can determine the transformation’s action on any vector in the space due to their role in expressing any vector as a linear combination of basis vectors.

Knowing how transformations affect these standard vectors helps simplify computations and gain insights into the behavior of more complex vectors under the same transformations.

Image Under Transformation

Understanding the image under transformation is key to visualizing what linear transformations actually do. For every basis vector we transformed in the exercise, the result was an 'image' vector. This concept of an image is crucial:

• The image \( A\bold{e}_1 \) tells us how \( \bold{e}_1 \) is represented in the transformed space, resulting in \( \begin{bmatrix} 2 \ 3 \end{bmatrix} \).
• Similarly, \( A\bold{e}_2 = \begin{bmatrix} -1 \ 1 \end{bmatrix} \) and \( A\bold{e}_3 = \begin{bmatrix} 5 \ -2 \end{bmatrix} \).
• Each 'image' vector reflects how space is stretched, compressed, or rotated by the transformation.

When these are combined appropriately, they determine the image of any vector in the original space. The image of a whole space under transformation maps out a new shape or position in another space. That's why understanding individual image vectors is fundamental—they illustrate the matrix's effect, demonstrating everything from dimensionality reduction to vector alignment in linear transformations.