(Solved): Find the standard matrix of the linear transformation \( T: \mathbb{R}^{2} \rightarrow \mathbb{R}^ ...

Find the standard matrix of the linear transformation \( T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \), where \( T \) first performs a horizontal shear that maps \( \mathbf{e}_{2} \) into \( \mathbf{e}_{2}-4 \mathbf{e}_{1} \) but leaves \( \mathbf{e}_{1} \) unchanged, and then reflects the result through the origin. \[ \left[\begin{array}{cc} 1 & 4 \\ 0 & -1 \end{array}\right] \] \[ \left[\begin{array}{cc} -1 & 4 \\ 0 & -1 \end{array}\right] \] \[ \left[\begin{array}{cc} -1 & 0 \\ 4 & -1 \end{array}\right] \] \[ \left[\begin{array}{cc} -1 & -4 \\ 0 & -1 \end{array}\right] \] \[ \left[\begin{array}{cc} -1 & 0 \\ 4 & 1 \end{array}\right] \]