91Ó°ÊÓ

A system of linear equations consists of multiple linear equations involving the same set of variables. In this problem, we have the equations \( u = x + 2y \) and \( v = x - y \). These equations define relationships among the variables \( x \), \( y \), \( u \), and \( v \). Our task is to solve for \( x \) and \( y \) in terms of \( u \) and \( v \). This is achieved by expressing \( x \) and \( y \) as functions of \( u \) and \( v \), which involves substitution and simplification methods. By solving, we find:

• \( x = \frac{u + 2v}{3} \)
• \( y = \frac{u - v}{3} \)

This shows how changes in \( u \) and \( v \) influence \( x \) and \( y \). Recognizing these relationships is essential for understanding solutions to linear systems.

Transformation of coordinates involves mapping points from one coordinate system to another. It essentially means changing the way we describe points in the plane with respect to the frame of reference. In our given exercise, we're applying a transformation defined by \( u = x + 2y \) and \( v = x - y \). This changes points described in terms of \( x \) and \( y \) to a new system using \( u \) and \( v \).Conversion involves plugging the coordinates into the transformation equations. For example, the intersection points such as \((0, 0)\), \((2, 0)\), and \( (\frac{2}{3}, \frac{2}{3}) \) in the original plane are re-mapped to new points in the \( uv \)-plane, illustrating how the shape and position of regions evolves under transformation. This process helps us visualize how geometric and algebraic properties of shapes alter when subjected to different coordinate frameworks.

Determinants play a crucial role in linear algebra, especially in understanding transformations. They provide information about the scaling and orientation effects of a transformation. In this exercise, we use the determinant of the Jacobian matrix to compute \( \frac{\partial(x, y)}{\partial(u, v)} \). This Jacobian determinant resolves the overall effect of changing variables from \((x, y)\) to \((u, v)\).The determinant is given by:\[\det\begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix}\]Computing this determinant gives us \( -\frac{1}{3} \), showing that the transformation reduces the area of regions in the \( xy \)-plane by \( \frac{1}{3} \) when mapped to the \( uv \)-plane. Understanding this helps in assessing changes in measurements and other properties during transformations.

Partial derivatives deal with the rate of change of multivariable functions concerning one variable while keeping others constant. When exploring transformations like the ones in our problem, partial derivatives help us understand how small changes in one variable affect the others.In the context of our transformation, we consider the partial derivatives:

• \( \frac{\partial x}{\partial u} = \frac{1}{3} \)
• \( \frac{\partial x}{\partial v} = \frac{2}{3} \)
• \( \frac{\partial y}{\partial u} = \frac{1}{3} \)
• \( \frac{\partial y}{\partial v} = -\frac{1}{3} \)

These derivatives show how \( x \) and \( y \) individually react to changes in \( u \) and \( v \). Calculating these derivatives is a fundamental step in forming the Jacobian matrix, which is used to analyze the effects of transformations on areas and orientations.