Question

in the following input and output grids, $\mathbf{a}$ maps the circle on the left to the oval on the right:

Answer 1

The transformation represented by matrix $\mathbf{a}$ stretches the circular shape on the left grid into the oval shape on the right grid.

Step-by-step explanation:

Matrix transformations are geometric operations performed on shapes using matrices. In this case, matrix $\mathbf{a}$ is responsible for the morphing of the circular shape into the oval shape. The matrix $\mathbf{a}$ operates on the coordinates of the points in the circular shape to yield the transformed oval shape.

To understand the specific transformation, let's consider a 2x2 matrix for this 2D transformation. If $\mathbf{a}$ is the transformation matrix, and $\mathbf{P}$ represents a point in the circular shape, the transformed point $\mathbf{P'}$ is given by the equation $\mathbf{P'} = \mathbf{aP}$. Each point's coordinates undergo multiplication by the matrix $\mathbf{a}$ to obtain the new coordinates, resulting in the oval shape.

The elements of matrix $\mathbf{a}$ determine the nature of the transformation: scaling, rotation, shearing, etc. In this case, the matrix $\mathbf{a}$ is primarily responsible for scaling along specific axes, causing the circular shape to elongate into an oval shape. The transformation matrix's elements directly affect the degree and direction of stretching or compressing, leading to the observed morphing from a circle to an oval in the grids.

Understanding how matrices operate on shapes helps comprehend the manipulation of points to achieve various transformations, crucial in fields like computer graphics, physics, and engineering for modeling and visual representation of objects and systems.

Here is complete question;

"In the following input and output grids, the transformation represented by matrix $\mathbf{a}$ morphs the circular shape on the left grid into the oval shape on the right grid."

Answer 2

The subject of this question is Mathematics and it involves mapping a circle to an oval using coordinates on a grid.

Step-by-step explanation:

The subject of this question is Mathematics. The question involves mapping a circle to an oval using coordinates on a grid.

For example, if we have a circle with coordinates (0,2) and (3,8) on the left and an oval with coordinates (0,2) and (3,2) on the right, we can say that the transformation from the circle to the oval is represented by the function a(x,y) = (x,2). This means that for any point (x,y) on the left circle, we map it to the point (x,2) on the right oval.