Problem 38 - Calculus for Biology and Medicine

Question

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The map represents a 90-degree counterclockwise rotation about the origin.

Step 1: Understand the Linear Transformation

We are trying to interpret the linear transformation defined by the matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \). This transformation maps a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) in the plane to \( \mathbf{y} = A\mathbf{x} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \).

Step 2: Perform the Matrix Multiplication

Multiply the matrix \( A \) by the vector \( \mathbf{x} \).\[A\mathbf{x} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} (0)(x_1) + (1)(x_2) \ (-1)(x_1) + (0)(x_2) \end{bmatrix} = \begin{bmatrix} x_2 \ -x_1 \end{bmatrix}\]

Step 3: Identify the Geometric Transformation

The result \( \begin{bmatrix} x_2 \ -x_1 \end{bmatrix} \) represents the vector \( \mathbf{x} \) rotated by 90 degrees counterclockwise around the origin. This is because the transformation swaps the components of the vector and changes the sign of the first component.

Step 4: Conclusion

The map \( \mathbf{x} \mapsto A\mathbf{x} \) given by the matrix \( A \) represents a 90-degree counterclockwise rotation about the origin in the cartesian plane.

Key Concepts

Matrix Multiplication

Matrix multiplication is a key step in understanding linear transformations like the one defined by the matrix\( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \). This operation involves combining a matrix with a vector to produce a new vector. Each element of the new vector is calculated as a sum of the products of the elements of the rows of the matrix with the columns of the vector. For our given matrix and vector, the operation looks like this:

• For the first row of the matrix, compute \((0)(x_1) + (1)(x_2) = x_2\).
• For the second row of the matrix, compute \((-1)(x_1) + (0)(x_2) = -x_1\).

These computations result in the vector \( \begin{bmatrix} x_2 \ -x_1 \end{bmatrix} \). This process is simple yet powerful, allowing us to transform vectors and analyze their properties in new ways.

Geometric Interpretation

The geometric interpretation of a linear transformation often involves visualizing how vectors are affected by the transformation. In this case, when the matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) is applied to any vector \( \mathbf{x} \), it results in the vector \( \begin{bmatrix} x_2 \ -x_1 \end{bmatrix} \). This transformation swaps the original components of the vector and changes the sign of the first, effectively rotating the vector.

Understanding Rotation in 2D

In two-dimensional space, this matrix operation translates to a rotation. Specifically, this operation results in rotating the vector 90 degrees counterclockwise around the origin. The swap of components and sign change are visual indicators of this geometric transformation. Such insights are crucial for applications in graphics, physics, and more where transformations are used to simulate motion or perspective.

Rotation Matrix

A rotation matrix is a specific type of linear transformation matrix used to rotate vectors around the origin of a coordinate system. Our matrix, \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \), is one of the simplest rotation matrices, representing a 90-degree rotation.

Characteristics of Rotation Matrices

Rotation matrices have several important properties:

• They are orthogonal, meaning their transpose is also their inverse.
• They preserve the length of vectors, as rotations do not change vector magnitudes.
• They alter only the direction of vectors, not their amplitude.
• The determinant of a 2D rotation matrix is always 1 or -1, indicating a pure rotation with no scaling.

Understanding these properties helps in comprehending how rotation matrices work and are applied in real-world scenarios. Whenever a 90-degree rotation is needed in mathematics or physics, this matrix comes into play effortlessly performing the transformation.