Give an example of a vector space and a subset of the set of vectors in it such that the subset together with the axpy operation is not a vector space.
$\mathcal V=\mathbb R^2$, $\left(\{(1,1)\},\circ\right)$ is not a vector space.
Prove $\mathcal {\dim(V\oplus W)=\dim V+\dim W-\dim (V\cap W)}$. Hint: Use of basis sets makes the details easier. $\mathcal {V\oplus W\equiv V\cup W}$
Let $\{v_1,\dots,v_k\}$ be a basis for $\mathcal {V\cap W}$. Extend it to form a basis for $\mathcal V$: $\{v_1,\dots,v_k,v_{k+1},\dots,v_n\}$ with $\dim\mathcal V=n$, and do the same for $\mathcal W$ with $\dim \mathcal W=m$, getting $\{v_1,\dots,v_k,w_{k+1},\dots,w_m\}$. This is possible since $\mathcal {V\cap W\subseteq V}$ and $\mathcal {\dim (V\cap W)\le\dim V}$ and also for $\mathcal W$. * Since $\mathcal V\oplus W=\{v+w:v\in\mathcal V,w\in\mathcal W\}$, any vector in it can be written by linear combination of $\{v_1,\dots,v_k,v_{k+1},\dots,v_n,w_{k+1},\dots,w_m\}$ and no less; since these are linearly independent they form a basis for $V\oplus W$. So $n+m-k=n+m-k$.
If $\mathcal {V_1,\dots,V_n}$ are disjoint vector spaces prove $\mathcal{\dim V=\sum_{i=1}^n \dim V_i}$
Apply what we just proved to disjoint vector spaces $\mathcal {V_1,V_2}$ to get $\dim (\mathcal {V_1\oplus V_2})=\dim \mathcal V_1+\dim \mathcal V_2-0$. Then use induction.
Could someone phrase the asterisked part * in a more logical way, if it can?
