Problem 21 - Differential Equations and Linear Algebra

Question

Determine the linear transformation \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) that has the given matrix. $$A=\left[\begin{array}{rrr} 2 & 2 & -3 \\ 4 & -1 & 2 \\ 5 & 7 & -8 \end{array}\right]$$.

Step-by-Step Solution

Answer

The linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) with the given matrix \(A=\left[\begin{array}{rrr} 2 & 2 & -3 \\\ 4 & -1 & 2 \\\ 5 & 7 & -8 \end{array}\right]\) can be represented as: $$T(x) = \begin{bmatrix} 2x_1 + 2x_2 - 3x_3 \\ 4x_1 - x_2 + 2x_3 \\ 5x_1 + 7x_2 - 8x_3 \end{bmatrix}$$ where \(x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) is a vector in \(\mathbb{R}^3\).

Step 1: Determine the dimensions of matrix A

Observe the given matrix $$A=\left[\begin{array}{rrr} 2 & 2 & -3 \\\ 4 & -1 & 2 \\\ 5 & 7 & -8 \end{array}\right]$$. There are 3 rows and 3 columns, which means the matrix A is a 3x3 matrix. Since the transformation is from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), we have \(n=3\) and \(m=3\).

Step 2: Write down the transformation using the given matrix A

Now that we know the dimensions of the matrix A, we can write down the transformation. Let \(x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\) be a vector in \(\mathbb{R}^3\). The linear transformation T can be represented as: $$T(x) = Ax = \left[\begin{array}{rrr} 2 & 2 & -3 \\\ 4 & -1 & 2 \\\ 5 & 7 & -8 \end{array}\right]x = \begin{bmatrix} 2x_1 + 2x_2 - 3x_3 \\ 4x_1 - x_2 + 2x_3 \\ 5x_1 + 7x_2 - 8x_3 \end{bmatrix}$$ Now, we have the linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) expressed in the form of the given matrix \(A\).

Key Concepts

matrix representation

Matrix representation is a way to express linear transformations using matrices. This offers a compact and powerful means to apply linear changes to vectors. (
) To represent a linear transformation using matrices, we consider both the input and the output spaces. The matrix provides a recipe for converting input vectors into output vectors. (
) In the exercise, the matrix \( A \) is \(3 \times 3\) with elements populated in a specific order. Each element in the matrix signifies how much of each component of the input vector will contribute to forming each component of the output vector. (
) A linear transformation \( T \) from \( \mathbb{R}^n \) to \( \mathbb{R}^m \) is represented by an \( m \times n \) matrix, where the matrix essentially acts on input vectors by matrix-vector multiplication.

dimensions of matrix

Understanding matrix dimensions is key when working with transformations. A matrix's dimensions are typically noted as rows by columns. (
) In the case of transformation matrices, these dimensions directly affect how the transformation operates on vectors. (
) For our matrix \( A \) which is \( 3 \times 3 \), this means:

• There are 3 rows, indicating that 3 output components can be produced for any input vector.
• There are 3 columns, implying any input vector must have 3 components.

This dimension consistency is crucial. The number of columns in the matrix must match the number of entries in the vector. This ensures that every aspect of the input can be processed into the output. Both input and output dimensions for \( T(x) = Ax \) line up because the matrix has dimensions that fit the spaces \( \mathbb{R}^3 \rightarrow \mathbb{R}^3 \). (
) Understanding these dimensions helps us ensure transformations are applied correctly.

vector transformation

Vector transformation involves converting an input vector from one space to another using a linear transformation. In a transformation \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \) represented by matrix \( A \), vectors from \( \mathbb{R}^n \) are transformed into vectors in \( \mathbb{R}^m \). (
) The process utilizes matrix-vector multiplication, which efficiently computes the new vector. For example, given vector \( x = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \), the transformation \( T(x) = Ax \) is realized by computing:

• The first element of the new vector is \( 2x_1 + 2x_2 - 3x_3 \).
• The second element is \( 4x_1 - x_2 + 2x_3 \).
• The third element becomes \( 5x_1 + 7x_2 - 8x_3 \).

This transformation process applies consistently for any vector from the input space, using the defined matrix to yield reliable results.