Problem 3 - Differential Equations and Linear Algebra

Question

For the transformation of \(\mathbb{R}^{2}\) with the given matrix, sketch the transform of the square with vertices \((1,1),(2,1),(2,2),\) and (1,2). $$A=\left[\begin{array}{rr}-2 & -2 \\\\-2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

After the transformation of the given square using the matrix \(A = \begin{bmatrix}-2 & -2\\-2 & 0\end{bmatrix}\), the new vertices of the transformed square are (-4, -2), (-6, -4), (-8, -4), and (-6, -2). Plot these transformed vertices on a coordinate plane to sketch the transformed square.

Step 1: Transform each vertex of the square

To apply the transformation to the square vertices, we will multiply the given matrix, A, by each vertex as a column vector. The vertices of the square are (1,1), (2,1), (2,2) and (1,2), so we will need to perform the following matrix multiplications: 1. A × (1,1) 2. A × (2,1) 3. A × (2,2) 4. A × (1,2)

Step 2: Compute the transformed vertices

Now, let us compute the transformed vertices by performing the matrix multiplications: 1. A × (1,1) = \(\begin{bmatrix}-2 & -2\\-2 & 0\end{bmatrix}\) × \(\begin{bmatrix}1\\1\end{bmatrix}\) = \(\begin{bmatrix} -4 \\ -2\end{bmatrix}\) 2. A × (2,1) = \(\begin{bmatrix}-2 & -2\\-2 & 0\end{bmatrix}\) × \(\begin{bmatrix}2\\1\end{bmatrix}\) = \(\begin{bmatrix} -6 \\ -4\end{bmatrix}\) 3. A × (2,2) = \(\begin{bmatrix}-2 & -2\\-2 & 0\end{bmatrix}\) × \(\begin{bmatrix}2\\2\end{bmatrix}\) = \(\begin{bmatrix} -8 \\ -4\end{bmatrix}\) 4. A × (1,2) = \(\begin{bmatrix}-2 & -2\\-2 & 0\end{bmatrix}\) × \(\begin{bmatrix}1\\2\end{bmatrix}\) = \(\begin{bmatrix} -6 \\ -2\end{bmatrix}\) So, the transformed vertices are: (-4, -2), (-6, -4), (-8, -4), and (-6, -2).

Step 3: Sketch the original and transformed squares

Finally, we will sketch the original square and the transformed square. The original vertices are (1,1), (2,1), (2,2), and (1,2). The transformed vertices are (-4, -2), (-6, -4), (-8, -4), and (-6, -2). Draw the squares using these vertices on a coordinate plane and label each vertex with its coordinates.

Key Concepts

Matrix Multiplication

Matrix multiplication is crucial for transforming points in coordinate geometry. In our exercise, we use the transformation matrix \\( A = \begin{bmatrix} -2 & -2 \ -2 & 0 \end{bmatrix} \) to transform vertices of a square in \( \mathbb{R}^2 \). Each vertex of the square, represented as a column vector, is multiplied by matrix A. This means for a vector \( \begin{bmatrix} x \ y \end{bmatrix} \), the transformation is computed by performing dot products:

• Row 1 of A: \\( (-2)x + (-2)y \)
• Row 2 of A: \\( (-2)x + 0y \)

Thus, matrix multiplication here directly modifies the spatial properties of the shape, resulting in new coordinates for each vertex.
By systematically applying the multiplication for each vertex, we deduce the new shape's placement in the plane.

Vector Transformation

Vector transformations involve altering the position and orientation of vectors on a plane through matrix transformations. In this exercise, each vertex of the square acts as a vector that we transform. For instance, the vertex (1,1) is transformed by multiplying it with the matrix \\( A = \begin{bmatrix} -2 & -2 \ -2 & 0 \end{bmatrix} \), resulting in a new vector \( \begin{bmatrix} -4 \ -2 \end{bmatrix} \).
Transformations can also entail scaling, rotating, or reflecting the vector. However, in this example, the transformation matrix scales and skews the vectors, altering the direction and length according to its elements. This is a fundamental operation in both linear algebra and computer graphics, as it allows for systematic manipulation of geometric figures.

Coordinate Geometry

Coordinate geometry provides a framework to analyze the transformations of shapes using their vertex coordinates. Initially, the square in our exercise is defined by vertices (1,1), (2,1), (2,2), and (1,2). When these vertices are transformed, it results in new coordinates: (-4, -2), (-6, -4), (-8, -4), and (-6, -2).

• Each transformed point can be visualized clearly on a coordinate plane.
• This helps in understanding how transformations affect geometric figures.

By placing the transformed points accurately and connecting them, we can see the effect of the matrix on the square, emphasizing how algebraic transformations translate into geometric shifts.

Sketching Transformations

Sketching transformations enables visual understanding of how matrices affect shapes like squares and rectangles. First, the original shape with vertices (1,1), (2,1), (2,2), and (1,2) is drawn. Then, using the transformed vertices (-4, -2), (-6, -4), (-8, -4), and (-6, -2), sketch the transformed shape.
By sketching:

• You'll see how points in the original shape correspond to points in the transformed one.
• It highlights changes in position, orientation, and size.

Sketching serves as a practical demonstration of theoretical concepts like vector transformation and matrix multiplication. It provides a tangible link between abstract calculations and real-world geometrical interpretation.