I am in the process of teaching some first year students about subspaces of $\mathbb{R}^{n}$ and need to stress that to check that $V\subseteq\mathbb{R}^{n}$ is a subspace, then you need to show that it is non empty. Specifically, they know that a subset $V\subseteq \mathbb{R}^{n}$ is a subspace if and only if the following three conditions hold:
- $V\neq\emptyset$;
- If $x,y\in V$, then $(x+y)\in V$; and
- If $x\in V$ and $\lambda\in\mathbb{R}$, then $\lambda\cdot x\in V$.
Is there actually an example of a subset $\emptyset=V\subseteq\mathbb{R}^{n}$ that is defined by a set of conditions such that if $x$ and $y$ were to satisfy those conditions, then $(x+y)$ and $\lambda\cdot x$ would also satisfy those conditions?
Edit: So specifically, has anyone got an example of a set of conditions on the coefficients of vectors in $\mathbb{R}^{n}$ such that
$V=\left\{\left(\begin{array}{c} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{array}\right):a_{1},\ldots,a_{n}\text{ satisfy certain conditions}\right\}=\emptyset$
but that if $(a_{1},\ldots,a_{n})^{T}$ and $(b_{1},\ldots,b_{n})^{T}$ were to satisfy the conditions, then so would $a_{1}+b_{1}$ etc. So basically I want a set of conditions that are closed under addition and scalare multiplication, but together are inconsistent.
For example, $3\vert a_{1}$ is a condition that is closed under addition but not under scalar multiplication, $\vert a_{1}\vert \geq \vert a_{2}\vert$ is closed under scalar multiplication but not addition etc.
I hope that clarifies things a bit.
Best Answer
I think the problem lies in the fact the zero vector is always an element of a vector space.