Problem 12 - Differential Equations and Linear Algebra

Question

Determine a formula for the linear transformation meeting the given conditions. \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{5}\) given by \(T(\mathbf{x})=A \mathbf{x},\) where $$A=\left[\begin{array}{rr} -1 & 4 \\ 0 & 2 \\ 3 & -3 \\ 3 & -3 \\ 2 & -6 \end{array}\right]$$

Step-by-Step Solution

Answer

The formula for the linear transformation T: \(\mathbb{R}^{2} \rightarrow \mathbb{R}^{5}\) with the given transformation matrix is: \[T(\mathbf{x}) = \left[\begin{array}{c} -x_1 + 4x_2 \\ 2x_2 \\ 3x_1 - 3x_2 \\ 3x_1 - 3x_2 \\ 2x_1 - 6x_2 \end{array}\right]\]

Step 1: Define the transformation matrix A and a general vector in the domain

First, rewrite the given transformation matrix A as: \[A = \left[\begin{array}{rr} -1 & 4 \\ 0 & 2 \\ 3 & -3 \\ 3 & -3 \\ 2 & -6 \end{array}\right]\] Now, let's consider a general vector \(\mathbf{x} \in \mathbb{R}^{2}\) as \(\mathbf{x} = \left(\begin{array}{c} x_{1} \\ x_{2} \end{array}\right)\). The goal is to find an expression for the image of this vector under the transformation T, i.e., \(T(\mathbf{x})\).

Step 2: Perform the matrix-vector multiplication

Now, we will multiply A by the vector \(\mathbf{x}\) to find the resulting vector. Recall that matrix-vector multiplication is performed by taking the dot product of each row of the matrix with the vector: \(T(\mathbf{x}) = A \mathbf{x} = \left[\begin{array}{rr} -1 & 4 \\ 0 & 2 \\ 3 & -3 \\ 3 & -3 \\ 2 & -6 \end{array}\right] \left(\begin{array}{c} x_{1} \\ x_{2} \end{array}\right)\) To compute the entries of the resulting vector, let's take the dot product of each row of A with \(\mathbf{x}\): \[\begin{aligned} T(\mathbf{x}) &= \left[\begin{array}{c} (-1)(x_1) + 4(x_2) \\ 0(x_1) + 2(x_2) \\ 3(x_1) - 3(x_2) \\ 3(x_1) - 3(x_2) \\ 2(x_1) - 6(x_2) \end{array}\right] \\ &= \left[\begin{array}{c} -x_1 + 4x_2 \\ 2x_2 \\ 3x_1 - 3x_2 \\ 3x_1 - 3x_2 \\ 2x_1 - 6x_2 \end{array}\right] \end{aligned}\]

Step 3: Write the formula for the linear transformation T

Now we have an explicit formula for the transformation T as a function of the input vector \(\mathbf{x}\): \[T(\mathbf{x}) = \left[\begin{array}{c} -x_1 + 4x_2 \\ 2x_2 \\ 3x_1 - 3x_2 \\ 3x_1 - 3x_2 \\ 2x_1 - 6x_2 \end{array}\right]\] This gives us the desired formula for the linear transformation T, which maps any vector \(\mathbf{x} \in \mathbb{R}^{2}\) to a vector in \(\mathbb{R}^{5}\).

Key Concepts

Matrix-Vector Multiplication

Matrix-vector multiplication is a fundamental operation where a matrix, which represents a linear map, is applied to a vector, which can be thought of as an input point or space. In our context, picture the transformation matrix 'A' as a set of instructions that will convert a two-dimensional vector into a five-dimensional vector.

The way this works is that each element of the resulting vector is calculated as the dot product of a row of the matrix with the entire vector. To visualize it, imagine that each row in the matrix is a gatekeeper, modifying the input vector incrementally before passing it on. The first component of the output vector is the product of the first row of the matrix and the original vector, the second component is the product of the second row, and so on, until all five components of our transformed vector in \( \mathbb{R}^{5} \) are determined.

Transformation Matrix

The concept of a transformation matrix is at the heart of understanding linear transformations. A transformation matrix is an array of numbers (matrix) that, when applied to vectors, changes them systematically. In technical terms, it \'represents\' the linear map between two vector spaces. In our exercise, the matrix A acts as such a transformation matrix, mapping two-dimensional vectors from \( \mathbb{R}^{2} \) to five-dimensional vectors in \( \mathbb{R}^{5} \).

When you see a matrix that operates as a transformation matrix, you can think of it as a tool that reshapes, rotates, scales, or does any kind of linear manipulation to the space of the vectors it is applied to. The number of rows indicates the dimension of the space where the vectors are sent, while the number of columns reflects the dimension of the space from where the vectors are coming.

Vector Space

A vector space is an algebraic structure that is composed of vectors, which can be added together and multiplied by scalars, that is, numbers. Vector spaces are the playgrounds where linear transformations occur. In our problem, we deal with the vector space \( \mathbb{R}^{2} \) composed of all two-dimensional vectors, which is the domain, and \( \mathbb{R}^{5} \) the codomain or the space to which our vectors are sent.

Understanding the structure and rules of vector spaces is essential for comprehending linear algebra. They are defined by a set of axioms that allow for vector addition and scalar multiplication. Each vector space has a zero vector, which is the equivalent of the number zero in the fields of real numbers—it's the vector that, when added to any other vector, doesn't change the other vector.

Dot Product

The dot product is a way of multiplying two vectors to yield a scalar. It is instrumental in matrix operations, including matrix-vector multiplication. When you take a dot product, you're essentially measuring how much one vector extends in the direction of another.

During matrix-vector multiplication, each entry of the resultant vector is computed by taking the dot product of a row of the matrix with the input vector. Conceptually, this fuses the elements of the vector with the individual characteristics of each component it interacts with in the matrix. It’s comparable to how ingredients combine differently to make unique recipes, and the dot product ensures that the ingredients (vector components) mix in a specific and consistent way.