(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let $$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$ Determine the dimension of $V / W$. (B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by $$\begin{aligned}
& T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\
& T e _ { 2 } = e _ { 3 } \\
& T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } .
\end{aligned}$$ Determine the nullity of $T$.
(A) (5 marks) Let $n \geq 2$ be an integer. Let $V$ be the $\mathbb { R }$-vector-space of homogeneous real polynomials in three variables $X , Y , Z$ of degree $n$. Let $p = ( 1,0,0 )$. Let
$$W = \left\{ f \in V \left\lvert \, f ( p ) = \frac { \partial f } { \partial X } ( p ) \right. \right\}$$
Determine the dimension of $V / W$.\\
(B) (5 marks) A linear transformation $T : \mathbb { R } ^ { 9 } \longrightarrow \mathbb { R } ^ { 9 }$ is defined on the standard basis $e _ { 1 } , \ldots , e _ { 9 }$ by
$$\begin{aligned}
& T e _ { i } = e _ { i - 1 } , \quad i = 3 , \ldots , 9 \\
& T e _ { 2 } = e _ { 3 } \\
& T e _ { 1 } = e _ { 1 } + e _ { 3 } + e _ { 8 } .
\end{aligned}$$
Determine the nullity of $T$.