4. Answer the following questions about Euclidean subspaces.
(a) Consider the following subsets of Euclidean space $\mathbb{R}^4$ defined by
$U = \left\{ \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \mid 6y + 5z = 3w - 4x \right\}$ and $W = \left\{ \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \mid 6x^2y + 5wz = 0 \right\}$
Without writing a proof, explain why only one of these subsets is likely to be a subspace.
(b) Consider the following subset of Euclidean space $\mathbb{R}^3$
$Q = \left\{ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \mid y = 5xy + z \right\}$
Prove that $Q$ is not a subspace.
(c) Consider the following subset of Euclidean space $\mathbb{R}^3$
$R = \left\{ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \mid y - 4z = -2x \right\}$
Prove that $R$ is a subspace.
