Which of the following are subspace in $\mathbb{R}^4$? Which of the following are subspaces in $\Bbb R^4$?
(a) $\mathrm{span}([1,2,-1-2],[0,2,1,0],[3,1,1,2],[1,2,1,3])$
(b) $\mathrm{span}([1,2,1,2,1],[2,1,2,3,5],[-1,1,0,0,1],[2,1,3,4,2])$
(c) The hyper plane in  $\Bbb R^4$ given by the equation $x+2y-z-3t=0$
(d) The hyper plane in  $\Bbb R^4$ given by the equation $x+2y-z-3t=2$
(e) The set of all answers to problem 7
(I got $v=[2-x,2,-1,x]$ as the answer for Question number 7)
My guess:
$a$, $b$, and $c$ are a subspace of $\mathbb{R}^4$ because the vectors in $a$ and $b$ are linearly independent and the equation in $c$ is set to $0$. $d$ is not a subspace because the equation is set to $2$. I have no clue about $e$.

(a) is a subspace since the span of any subset of a vectors space is a subspace
(b) is not a subspace of $\mathbb{R}^4$ since it consists of vectors from $\mathbb{R}^5$.
(c) is a subspace since it's the null space of a matrix
(d) is not a subspace since the zero vector is not an element
(e) is not a subspace since the zero vector is not an element
