91Ó°ÊÓ

An inverse transformation reverses the effect of an original transformation. In the context of the given exercise, we have a transformation expressed by equations \(u = -5x + 4y\) and \(v = 2x - 3y\), which map points from the \(xy\)-plane to the \(uv\)-plane. The task is to find the inverse transformation, denoted by \((x, y) = T^{-1}(u, v)\).

To find the inverse, we solve the system of linear equations for \(x\) and \(y\). This requires expressing these variables in terms of \(u\) and \(v\).

• First, write the transformation equations in matrix form.
• Then, find the determinant of the matrix.
• Finally, apply matrix algebra to compute the inverse matrix.

This process reveals the inverse formulas: \(x = \frac{-3u - 4v}{7}\) and \(y = \frac{-2u - 5v}{7}\). Once these expressions are obtained, they should yield the original \(u\) and \(v\) when substituted back, confirming the correctness of the inverse transformation.

Matrix algebra is a powerful tool to solve systems of linear equations. In this exercise, the original transformation from the \(xy\)-plane to the \(uv\)-plane can be expressed using a matrix equation:
\[\begin{bmatrix} -5 & 4 \2 & -3 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} u \ v \end{bmatrix}\]
By organizing equations into a matrix form, it becomes easier to perform algebraic operations necessary to find the inverse transformation. The key steps in matrix algebra for solving or inverting systems include:

• Calculating the determinant of the matrix.
• Computing the inverse of the matrix, if it exists.
• Multiplying the inverse matrix with the constant matrix \([u, v]^T\) to obtain the original variables \([x, y]^T\).

The matrix inverse is computed as:
\[(1/det)\begin{bmatrix} -3 & -4 \ -2 & -5 \end{bmatrix} \] where the determinant \(det = 7\). Here, the inverse matrix allows us to reverse the transformation by reconfiguring \(u\) and \(v\) back to \(x\) and \(y\).

Determinants play a crucial role in matrix algebra, especially when finding the inverse of a matrix. The determinant is a scalar value that can give us important information about the matrix, such as whether an inverse exists.

For the matrix in question, it is \[\begin{bmatrix} -5 & 4 \ 2 & -3 \end{bmatrix}\]
To calculate the determinant, use the formula:

• Multiply the top-left by the bottom-right: \((-5)(-3) = 15\).
• Multiply the top-right by the bottom-left: \((4)(2) = 8\).
• Subtract the second product from the first: \(15 - 8 = 7\).

This determinant value is non-zero, implying the matrix is invertible. A non-zero determinant confirms that the system of equations has a unique solution, which is essential for calculating the inverse transformation.

A system of linear equations consists of two or more linear equations with the same variables. In this exercise, the equations are: \(u = -5x + 4y\) and \(v = 2x - 3y\).

Solving a system of linear equations involves finding values for the variables that satisfy all equations involved. In the context of transformations, solving means expressing \(x\) and \(y\) in terms of \(u\) and \(v\). Some methods for solving these systems include:

• Substitution, where you express one variable in terms of another and substitute back.
• Elimination, which involves adding or subtracting equations to eliminate a variable.
• Using matrix algebra to simplify the process, especially when dealing with more complex systems.

By applying these techniques, you can uncover how transformations map points from one plane to another and eventually back, solving for the inverse transformation as necessary. This creates an understanding of how different values of \(u\) and \(v\) map onto the \(x\) and \(y\) coordinates, completing the transformation cycle.