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A linear transformation is a mathematical function that maps input vectors to output vectors in a linear way. A key property of linear transformations is that they preserve the operations of vector addition and scalar multiplication.
This means that if you have vectors \( extbf{u} \) and \( extbf{v} \), and a scalar \( c \), a linear transformation \( T \) will satisfy:

• \( T(\textbf{u} + \textbf{v}) = T(\textbf{u}) + T(\textbf{v}) \)
• \( T(c \cdot \textbf{u}) = c \cdot T(\textbf{u}) \)

In the problem at hand, the transformation function \( y_1 = x_1 - 2x_2 \) and \( y_2 = 4x_1 - 3x_2 \) maps the vector \((x_1, x_2)\) to \((y_1, y_2)\). This can be represented in matrix form as \( \begin{pmatrix} y_1 \ y_2 \end{pmatrix} = \begin{pmatrix} 1 & -2 \ 4 & -3 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \). The matrix \( \begin{pmatrix} 1 & -2 \ 4 & -3 \end{pmatrix} \) embodies the transformation rules.

Matrix inversion is the process of finding the matrix \( A^{-1} \) that, when multiplied with the matrix \( A \), results in the identity matrix.
The identity matrix acts like the number 1 in matrix multiplication. For an \( n \times n \) matrix \( A \), the identity matrix \( I \) is such that \( AI = IA = A \).The matrix inversion plays a crucial role in solving systems of linear equations, where the inverse matrix translates output back to input, providing a reverse transformation.To invert our transformation matrix \( A = \begin{pmatrix} 1 & -2 \ 4 & -3 \end{pmatrix} \), we apply the formula for 2x2 matrices:\[ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]For our matrix:- Determinant \( ad - bc = 1(-3) - (-2)(4) = 5 \) (we calculate this first)- Then, \( A^{-1} = \frac{1}{5} \begin{pmatrix} -3 & 2 \ -4 & 1 \end{pmatrix} \)Thus, \( A^{-1} = \begin{pmatrix} -\frac{3}{5} & \frac{2}{5} \ -\frac{4}{5} & \frac{1}{5} \end{pmatrix} \), which can be used to reverse the transformation.

The determinant of a matrix is a special number that can be calculated from its elements.
It provides important properties about the matrix.For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant \( \det(A) \) is computed as:\[ \det(A) = ad - bc \]The determinant can tell us a few things:

• If \( \det(A) = 0 \), the matrix is singular, meaning it doesn't have an inverse.
• If \( \det(A) eq 0 \), an inverse exists, and the system of equations has a unique solution.

For the matrix in our problem, \( A = \begin{pmatrix} 1 & -2 \ 4 & -3 \end{pmatrix} \), we found that \( \det(A) = 5 \). Therefore, the inverse transformation exists, as we computed earlier.The determinant is essential in the process of both finding inverses and understanding system solvability.