Prompt What matrix operations can you use to translate, dilate, reflect, and rotate figures in the coordinate plane?

Matrices can be used to represent linear transformations such as those that occur when two-dimensional or three-dimensional objects on a computer screen are moved, rotated, scaled (resized) or undergo some kind of deformation. Each transformation is represented by a single matrix. Matrices are particularly useful for representing transformations in the field of computer graphics, because each transformation can be represented in a consistent format that is easily represented as a computer data structure. In addition, the transformation represented by a matrix M can be undone by applying the inverse of the matrix. In other words, matrix M -1 reverses the transformation applied by matrix M. We will be looking here at some basic examples of using matrices to represent different kinds of transformations of two-dimensional objects. Let's start with simple reflection.Imagine a single point A that has the co-ordinates x = 3, y = 4 relative to the origin (0, 0).Supposing we wish to find the matrix that represents the reflection of any point (x, y) in the x-axis. The transformation involved here is one in which the coordinates of point (x, y) will be transformed from (x, y) to (x, -y). For this to happen, x does not change, but y must be negated. We can therefore achieve the required transformation by multiplying y by minus one (-1). Because we are dealing with a two-dimensional image, all points in the two-dimensional plane are represented by two variables (x and y). The transformations we apply to a point must set values for x and y independently. We will therefore represent our transformation using the two-by-two matrix M:

M = ab

cd

The original position of point A relative to the origin is given by a position vector (x, y) which we can represent using a one-by-two column matrix which we'll call matrix V. By convention, the point created by a transformation is called the image of the original point, and is often given the same label but with the addition of a tick mark (a single apostrophe). Thus A will become A'. The position of point A' relative to the origin after the transformation has been applied is given by the position vector (x, -y), which we can represent using a second one-by-two column matrix which we'll call matrix V '. We can find V ' as follows.

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a variable is any characteristics, number, or quantity that can be measured or counted. a variable may also be called a data item. hope i !