determine whether the following are linear transformations from r2 int

Question

Determine whether the following are linear transformations from R2 into R². If the map is a linear transformation, provide a proof that it is a linear transformation by verifying that (LT1) and (LT2) hold. If the map is a not linear transformation, state one of the properties of a linear transformation that does not hold (either (LT1) or (LT2)) and give a counterexample showing that the property fails. T\left(\left[\begin{array}{l}
x \\
y
\end{array}\right]\right)=\left[\begin{array}{l}
y-4 y \\
2 x+y
\end{array}\right] T\left(\left[\begin{array}{l}
x \\
y
\end{array}\right]\right)=\left[\begin{array}{c}
0 \\
x y
\end{array}\right] T(\mathbf{u})=\mathbf{u}-2 \operatorname{proj}_{\mathbf{n}} \mathbf{u} for any u E R². You may want to recall the formula for the projection of one vector onto another from Q13in Length, Distance, Angles and Orthogonality. ) Let n be a fixed, non-zero vector in R.