Let $W = \left\{ ( a , b , c , d ) \in \mathbb { R } ^ { 4 } \mid 3 a - b + 6 c = 0 \right\}$ and $T : \mathbb { R } ^ { 4 } \longrightarrow W$ be a linear map with $T ^ { 2 } = T$. Suppose $T$ is onto. Pick the correct statement(s) from below. (A) $T ( u + v ) = T ( u ) + v$ for all $u \in \mathbb { R } ^ { 4 } , v \in W$. (B) $\operatorname { ker } ( T - I )$ contains three linearly independent vectors. (C) $( 1,3,0,2 ) \in \operatorname { ker } ( T )$. (D) If $v _ { 1 } , v _ { 2 } \in \operatorname { ker } ( T )$ are nonzero, then $v _ { 1 } = c v _ { 2 }$ for some $c \in \mathbb { R }$.
Let $W = \left\{ ( a , b , c , d ) \in \mathbb { R } ^ { 4 } \mid 3 a - b + 6 c = 0 \right\}$ and $T : \mathbb { R } ^ { 4 } \longrightarrow W$ be a linear map with $T ^ { 2 } = T$. Suppose $T$ is onto. Pick the correct statement(s) from below.\\
(A) $T ( u + v ) = T ( u ) + v$ for all $u \in \mathbb { R } ^ { 4 } , v \in W$.\\
(B) $\operatorname { ker } ( T - I )$ contains three linearly independent vectors.\\
(C) $( 1,3,0,2 ) \in \operatorname { ker } ( T )$.\\
(D) If $v _ { 1 } , v _ { 2 } \in \operatorname { ker } ( T )$ are nonzero, then $v _ { 1 } = c v _ { 2 }$ for some $c \in \mathbb { R }$.