C30 Define the linear transformation

\[ T: C^3 -> C^4, \mathbf{T}\left[\begin{array} {rrr} x_1 \\ x_2 \\ x_3 \end{array}\right]=\left[\begin{array} {rrr} 2x_1 -x_2 +5x_3 \\ -4x_1 +2x_2 -10x_3 \end{array}\right]\]

Compute the preimages \[T^{-1} = \left[\begin{array} {rrr} 2 \\ 3 \end{array}\right] and \space T^{-1} = \left[\begin{array} {rrr} 4 \\ 8 \end{array}\right]\]

library(pracma)
m1 <- matrix(c(2, -1, 5, 
                  2, -4, 2, -10, 3), nrow = 2, byrow = TRUE)
print("Augmented Matrix T^-1 [2,3]: ")
## [1] "Augmented Matrix T^-1 [2,3]: "
print(m1)
##      [,1] [,2] [,3] [,4]
## [1,]    2   -1    5    2
## [2,]   -4    2  -10    3
rref(m1)
##      [,1] [,2] [,3] [,4]
## [1,]    1 -0.5  2.5    0
## [2,]    0  0.0  0.0    1
m2 <- matrix(c(2, -1, 5, 
               4, -4, 2, -10, -8), nrow = 2, byrow = TRUE)
print("Augmented Matrix T^-1 [4,8]: ")
## [1] "Augmented Matrix T^-1 [4,8]: "
print(m2)
##      [,1] [,2] [,3] [,4]
## [1,]    2   -1    5    4
## [2,]   -4    2  -10   -8
rref(m2)
##      [,1] [,2] [,3] [,4]
## [1,]    1 -0.5  2.5    2
## [2,]    0  0.0  0.0    0

For m1 there is no vector which is member of \(T^{-1}\) and below is the Preimage for m2

\[ \left[\begin{array} {rrr} 2 \\ 0 \\ 0 \end{array}\right] + x_2 \left[\begin{array} {rrr} 1/2 \\ 1 \\ 0 \end{array}\right] + x_3 \left[\begin{array} {rrr} -5/2 \\ 0 \\ 1 \end{array}\right] \]