If $\{u,v,w\}$ is a spanning subset of $V$ then so is $\{u+v, v+w, w+u\}$
133087
Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
140668
Union of $2$ set $A, B$ where $ A$ is a subset of $B$
150720
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
37119
Determine Which of the following Sets is a Vector Space
14576
$(U\cap W)^0 = U^0 + W^0$
45485
Dimension of a subset
149907
$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
22758
If $\operatorname{ran} F \subseteq \operatorname{dom} G$, then $\operatorname{dom}(G \circ F) = \operatorname{dom} F$.
82578
Show that if $G$ is $2-$connected, then for any two vertices $u$ and $v$ there exists a cycle $C$ such that $u, v \in V (C)$
41520
Prove that $L(V,W)$ forms a vector space
41080
If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$
58285
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
19396
For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?
122271
A subset of a transitive set is transitive
85239
Proof $dim(W)=dim(V)=n \implies W=V$