# Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4$ When this transformation is epimorphic i.e. what should i look for in the reduced form of matrix of this linear transformation. My reduced matrix is:

$\begin{pmatrix} 1 & 1 & 1 & 2 \\ 0 & t-1 & 0 & 1 \\ 0 & -1 & t-2 & -1 \end{pmatrix}$

It seems the following.

The transformation $\psi$ is ephimorphic iff rank of its matrix is maximal, that is $3$.

Buy elementary transormations, which does not change the rank, we can transform the matrix as follows.

$\begin{pmatrix} 1 & 1 & 1 & 2 \\ 0 & t-1 & 0 & 1 \\ 0 & -1 & t-2 & -1 \end{pmatrix}$

$\begin{pmatrix} 1 & {\mathbf 0} & {\mathbf 0} & {\mathbf 0}\\ 0 & t-1 & 0 & 1 \\ 0 & -1 & t-2 & -1 \end{pmatrix}$

$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & t-1 & 0 & 1 \\ 0 & {\mathbf t-2} & t-2 & {\mathbf 0} \end{pmatrix}$

$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & {\mathbf 0} & 0 & 1 \\ 0 & t-2 & t-2 & 0 \end{pmatrix}$

$\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \\ 0 & {\mathbf 0} & t-2 & 0 \end{pmatrix}$

That is the rank equals $3$ iff $t\ne 2$ .

Answer. The transformation $\psi$ is ephimorphic iff $t\ne 2$.