Answer
No
Work Step by Step
Put matrix $A$ in Matlab.
$A=[4,-7,3,7,5 ; 6,-8,5,12,-8 ;-7,10,-8,-9,14 ; 3,-5,4,2,-6 ;-5,6,-6,-7,3]$
Use command $\operatorname{rref}(A)$
It returns reduced row echelon form of $A$.
$\operatorname{rref}(A)=\left[\begin{array}{ccccc}1 & 0 & 0 & 5 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$
$A$ has 4 pivot columns so $T$ does not map $\mathbb{R}^{5}$ onto $\mathbb{R}^{5}$.
