When will span of vectors plus a constant form a vector space?

For set $W$ defined as:

$W$ is not a vector space because if $d = 0$ then $2d + 1 \neq 0$ so $0$ is not in $W$.

Note that the set above can be rewrited as linear combinations of vectors plus a constant vector: $\{(0,0,1,0) + b(1, 2, 0, 0)+ d(-5,0,2,1) |\ b, d\ real \}$

The question can be generalized as:

If $W$ is the set of linear combinations of a set of vectors $\{v_1,...v_n \}$ plus a vector $v_0$ (all vectors are from $\mathbb{R}^m$), i.e.
$W=\{v_0+c_1v_1+c_2v_2+...+c_nv_n|\ v_0\neq 0\ ,\ c_i \text{ is any real scalar}\}$, under what conditions of $\{v_1,...v_n \}$ and $v_0$ will $W$ be a vector space?

$W$ can be a vector space though. For example, when $W = \{x+1|\ x \in \mathbb{R} \}$, $W = \mathbb{R}$ and $W$ is a vector space.

Solution 1:

Let $V$ be a vector space, $W\subseteq V$ a subspace and $v\in V$. Then $v+W$ is a subspace of $V$ iff $v\in W$.