The linear transformation \(R: M_{12} \rightarrow M_{21}\) is invertible, Determine a formula for the inverse linear transformation \(R^{-1}: M_{21} \rightarrow M_{12}\).
\[ R \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} a + 3b \\ 4a + 11b \end{pmatrix} \]
To find the formula for \(R^{-1}\), we need to undo the action of \(R\) by solving for \(a\) and \(b\) in terms of \(x\) and \(y\):
\[ a = a + 3b \] \[ b = 4a + 11b \]
Upon solving these equations, we find that \(a = b = 0\). Thus, the inverse linear transformation \(R^{-1}\) maps every element of \(M_{21}\) to the zero matrix in \(M_{12}\). So, the formula for \(R^{-1}\) is simply:
\[ R^{-1}(x, y) = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \]
This means that any element \((x, y)\) from \(M_{21}\) is mapped to the zero matrix \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\) in \(M_{12}\).
# Define the coefficient matrix
A <- matrix(c(1, -3, -4, 10), nrow = 2, byrow = TRUE)
# Define the right-hand side vector
b <- c(0, 0)
# Solve the system of equations
solution <- solve(A, b)
solution## [1] 0 0