## The Region R

The region R is a plane figure that is enclosed by a boundary. The boundary of the region R is the set of all points (x,y) such that x and y are both real numbers and x^2+y^2=1.

### The Boundary of R

The boundary of the region R is the set of points in the xyplane that are transformed to the uvplane by the given transformations. To sketch the image of the region R in the uvplane, we need to find the points on the boundary of R that are mapped to the uvplane.

### The Interior of R

To sketch the interior of R, we first need to find its boundaries. To do this, we note that the region is bounded by the x-axis, the y-axis, and the line y = x. We also see that it is unbounded above. Thus, we have the following sketch:

## The Transformation

The given transformation is a mapping from the Region R in the XY-Plane to the UV-Plane. The image of the Region R is the set of all points in the UV-Plane which are the images of points in the Region R.

### The Linear Transformation

A linear transformation is a transformation that preserves lines. In other words, if you have a line in the xy-plane, then the image of that line under a linear transformation will also be a line.

The linear transformation can be represented by a matrix. If you have a point (x, y), then the point (x’, y’) after the linear transformation can be found by multiplying the matrix by the vector:

$$\begin{pmatrix}a&b\c&d\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}ax+by\cx+dy\end{pmatrix}$$

Let’s look at an example to see how this works.

Example: Consider the linear transformation $$T(x, y) = \begin{pmatrix}2&1\1&2\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}.$$ The matrix for this linear transformation is:

$$A=\begin{pmatrix}2&1\1&2\end{pmatrix}.$$

To find the image of the point (1, 2) under this linear transformation, we simply multiply A by the vector [1 2] :

$$A\begin{pmatrix}1\2\end{pmatrix} = \begin{pmatrix}2&1\1&2\end{pmatrix}\begin{pmatrix}1\2\end{

### The Affine Transformation

A linear transformation is a transformation that can be represented by a matrix. An affine transformation is a linear transformation plus a translation. In other words, an affine transformation is a combination of a rotation, reflection, dilation, etc. (i.e. any linear transformation) plus a translation.

In 2 dimensions, an affine transformation can be represented by the following matrix:

$$\begin{bmatrix}a&b\c&d\end{bmatrix}$$

where $$a, b, c$$ and $$d$$ are real numbers. The number $$a$$ is called the scale factor in the $$x$$-direction, while $$d$$ is the scale factor in the $$y$$-direction. The numbers $$b$$ and $$c$$ represent horizontal and vertical shearing respectively. Note that this representation is only valid in 2 dimensions; in 3 dimensions or more, things are more complicated.

Let’s look at an example to see how this works in practice. Consider the following region in the XY-plane:

Now let’s say we want to sketch the image of this region after it has been transformed by the following matrix:

To do this, we need to apply the transformations described by each element of the matrix in turn to every point in our region. So first we’ll scale everything in the X-direction by a factor of 2:

Note that this has had the effect of doubling the lengths of all our vectors in the X-direction (i.e., they’ve been stretched). Now we’ll shear everything in the Y-direction by a factor of 0.5:

Shearing gives our vectors a slanted look; notice how every vector has been shifted horizontally or vertically (or both) by an amount that depends on its Y-coordinate (for horizontal shearing) or X-coordinate (for vertical shearing). Finally, we’ll translate everything up and to the right by 3 units each:

![](https://github

### The Nonlinear Transformation

The nonlinear transformation is a mapping from the xyplane to the uvplane that is not a function. In other words, there is more than one value of u for a given value of x, or more than one value of v for a given value of y. This type of transformation is not invertible, which means that it is not possible to determine the inverse mapping from the uvplane back to the xyplane.

## The Image

sketch the image of the region R in the xy-plane that is the result of applying the given transformations to the region R.

### The Boundary of the Image

The boundary of the image is the set of points (x,y) in the xyplane such that there exists a point (u,v) in the uvplane such that T(x,y)=(u,v).

### The Interior of the Image

The image of the region R in the xy-plane under the given transformations is the interior of the image of R in the UV-plane.