Problem 40 - Calculus for Biology and Medicine

Question

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\frac{1}{2}\left[\begin{array}{lr}\sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2}\end{array}\right]$$

Step-by-Step Solution

Answer

The transformation reflects over the line \( y = x \) and scales by \( \frac{1}{\sqrt{2}} \).

Step 1: Understand the Linear Transformation

The matrix \( A \) is a linear transformation that maps a vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \) in \( \mathbb{R}^2 \) to another vector \( \mathbf{y} = A\mathbf{x} \) by matrix multiplication. The transformation given by \( A \) can be expressed as \( A\mathbf{x} = \frac{1}{2} \begin{bmatrix} \sqrt{2} & -\sqrt{2} \ \sqrt{2} & \sqrt{2} \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \).

Step 2: Perform Matrix Multiplication

Compute the product \( A\mathbf{x} = \frac{1}{2} \begin{bmatrix} \sqrt{2}x_1 - \sqrt{2}x_2 \ \sqrt{2}x_1 + \sqrt{2}x_2 \end{bmatrix} \). The result shows how each vector \( \mathbf{x} \) is transformed by the matrix \( A \).

Step 3: Geometric Interpretation

The matrix \( A \) first reflects points over the line \( y = x \), and then scales them by \( \frac{1}{\sqrt{2}} \). The reflection is due to the combination of \( \sqrt{2} \) and \(-\sqrt{2} \) in different positions in the first row, which exchanges and modifies the components \( x_1 \) and \( x_2 \). Finally, multiplying by \( \frac{1}{2} \) results in a scaling of both the \( x \) and \( y \) components by \( \frac{1}{\sqrt{2}} \), given that \( \frac{1}{2} \sqrt{2} = \frac{1}{\sqrt{2}} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra used to transform vectors from one space to another. When performing matrix multiplication, start with a matrix and multiply it by a vector. The process involves taking the dot product of the rows of the matrix with the column vector. In this exercise, matrix \( A \) is multiplied by the vector \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \). The multiplication can be broken down into simple steps:

• Multiply the entries of each row of the matrix by the corresponding entries of the vector.
• Sum these products to form a new vector.

For the given matrix \( A = \frac{1}{2} \begin{bmatrix} \sqrt{2} & -\sqrt{2} \ \sqrt{2} & \sqrt{2} \end{bmatrix} \), this process yields \( A\mathbf{x} = \frac{1}{2} \begin{bmatrix} \sqrt{2}x_1 - \sqrt{2}x_2 \ \sqrt{2}x_1 + \sqrt{2}x_2 \end{bmatrix} \). This illustrates how matrix multiplication alters every vector \( \mathbf{x} \) differently depending on its components.

Geometric Interpretation

The geometric interpretation of a linear transformation provides a way to visualize how the transformation affects objects in space. For the matrix \( A \), each point originally at \( \mathbf{x} \) finds a new home at \( A\mathbf{x} \). This can be seen as a sequence of geometric actions. In this case:

• Reflection Over the Line: By forming a new vector, the matrix \( A \) reflects points across the line \( y = x \). This occurs due to the structure of the matrix where the combination of \( \sqrt{2} \) and \( -\sqrt{2} \) switches the original vector's coordinates and signs.
• Scaling: After the reflection, there's a uniform scaling by \( \frac{1}{\sqrt{2}} \). This shrinks the vector, maintaining the angles but reducing the length proportionally in all directions.

Understanding these actions helps in visualizing how the transformation changes shapes and trajectories in a two-dimensional space.

Reflection and Scaling

Reflection and scaling are crucial aspects of this particular linear transformation executed by the matrix \( A \). These actions modify the dimensions and orientation of vectors in a clear, tangible way.

• Reflection: Here, the reflection occurs due to the matrix’s determinant factors \( \sqrt{2} \) and \( -\sqrt{2} \), which essentially flips vectors over the line \( y = x \). It's like mirroring an image across a diagonal line, changing \( x_1 \) to \(-x_2 \) and vice versa.
• Scaling: This transformation does not distort the angles of the vector but uniformly scales them down by the factor \( \frac{1}{\sqrt{2}} \). This scaling is crucial for visualizing how the transformation distorts sizes without altering proportions. To visualize, imagine reducing a picture by keeping the aspect ratio the same.

Consider these modifications as the art of manipulating geometric figures, where reflection re-orients them, and scaling adjusts their size.