Caption for the picture.

Caption for the picture.

To compute the preimages we need to create an augmented matrix from the transformation and solution vectors

AT solution \[ \begin{bmatrix} -2 \\ 5 \\ 3 \end{bmatrix} \]

  • Augmented matrix

\[\begin{equation*} \mathbf{}\left[\begin{matrix} 1& -2& -1&-2 \\ 3&-1&2&5 \\ 1&1&2&3 \end{matrix}\right] \end{equation*} \]

Now row reduce the augmented matrix

## Warning: package 'pracma' was built under R version 3.4.4
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    1    0
## [2,]    0    1    1    0
## [3,]    0    0    0    1
  • As the pivot column is in the solution vector, the system is inconsistent , so the preimage is an empty set

At solution \[ \begin{bmatrix} -5 \\ 5 \\ 7 \end{bmatrix} \]

  • augmented matrix

\[\begin{equation*} \mathbf{}\left[\begin{matrix} 1& -2& -1&-5 \\ 3&-1&2&5 \\ 1&1&2&7 \end{matrix}\right] \end{equation*} \]

## Warning: package 'knitr' was built under R version 3.4.4

Now row reduce the augmented matrix

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    1    3
## [2,]    0    1    1    4
## [3,]    0    0    0    0

\[X = -1Z +3\] \[Y = -Z +4 \]

  • In Terms OF Z \[Z = -1X +3\]

\[Z = -1Y +4\]

  • PREIMAGE

\[\begin{equation*} \mathbf{}\left[\begin{matrix} 3 \\ 4 \\ 0 \end{matrix}\right]+ Z \left[\begin{matrix} -1 \\ -1 \\ 1 \end{matrix}\right] \end{equation*} \]