How to prove that set is a subspace? I am not able to prove the following theorem:-
A non-empty subset $W$ of$V$(a vector space) is a subspace if and only if for each $u$ and $v$ in $W$ and each scalar $c$ belongs to field of vector space $V$ which is set of real number $\mathbb{R}$, the sum $c u  + v\in W$.
I am not able to prove forward(the above stated theorem) and backward i.e., if $c u  +  v   \in  W$ then, $ u  and v$ belongs to $W$.
 A: You're misunderstanding how you should prove the converse direction.
Forward direction: if, for all $u,v\in W$ and all scalars $c$, $cu+v\in W$, then $W$ is a subspace
Backward direction: if $W$ is a subspace, then, for all $u,v\in W$ and all scalars $c$, $cu+v\in W$
Note that the “backward direction” follows easily from the definition of subspace. Note also that $cu+v\in W$ does not generally imply $u,v\in W$, even if $W$ is a subspace: think to $u\in V$ but $u\notin W$; then $(-1)u+u=0\in W$.
Hints for the proof of the “forward direction”: since $W$ is not empty, take $w\in W$ and consider $c=-1$, $u=w$ and $v=w$; then $cu+v=0\in W$ by assumption. Can you prove the rest?
You have to prove:


*

*$0\in W$

*for all $u,v\in W$, $u+v\in W$

*for all $u\in W$ and all scalars $c$, $cu\in W$


Fact 1 is covered by the hint.

 Fact 2 requires using $c=1$ in the assumed property.


 Fact 3: you can choose $v=0$ in the assumed property.

A: If
For $u,v \in W$ and $k\in F$ to prove that $u+v \in W$ and $ku \in F$.
$$\begin{array}{rcll}
u &\in& W & \text{assumtion} \\
v &\in& W & \text{assumtion} \\
u+v &\in& W & \text{set }c=1 \\
(k+1)u+v &\in& W & \text{set }c=k+1 \\
-(u+v)+((k+1)u+v) &\in& W & \text{set }c=-1\text{ and combine the two lines above} \\
ku &\in& W\\
\end{array}$$
Only if
For $u,v \in W$ and $c \in F$ to prove that $cu+v \in W$.
Since $W$ is a subspace, $cu \in W$ and hence $cu+v \in W$.

The "if" part reads "if for each $u,v \in W$ and $c \in F$ we have $cu+v \in W$, then $W$ is a subspace".
The "only if" part reads "if $W$ is a subspace, then for each $u,v \in W$ and $c \in F$ we have $cu+v \in W$.
