Is H is a Subspace of the Vector Space? 
So i figured out the answer to this problem, however i would appreciate if someone can explain each part and the thought process behind it. Im trying to understand it but my head isnt wrapping around it. Any help would be appreciated. Thanks.
 A: Let $H= \{ (x,y) \vert -2x-3y=6\}$ and we are trying to see if $H$ is indeed a subspace of $V= \mathbb{R}^2$.  This problem here is usually the general method to seeing if a given set is indeed a subspace of a larger vector space. 


*

*Is $H$ nonempty? If there is at least one solution to this linear equation then $H$ is clearly nonempty; given this can you find one such point in this set?

*Is $H$ closed under addition/ multiplication?  Notice that $(0,-2), (-3,0)$ are in the set. Is their sum, $(-3,-2)$ an element of $H$? 
Take either of the points mentioned and multiply by any scalar, is the result an element of $H$?


Just as a remark, recall that a subspace is also a vector space on its own, thus seeing that the origin is not contained in $H$ should raise a red flag. 
A: A good method to solve this kind of problems is to explicitly represent a vector of the subspace in question. In your case from $-2x-3y=6$ we find $y=-3-\frac{2}{3}x$ so the vectors in  $H$ have the form
$$
\begin{pmatrix}
x\\
-3-\frac{2}{3}x
\end{pmatrix} \quad \forall x\in \mathbb{R}
$$
now you can answer to the given questions. 
Clearly $H$ is not empty.
for the question 2) we have test with two vectors:
$$
\begin{pmatrix}
a\\
-3-\frac{2}{3}a
\end{pmatrix}+
\begin{pmatrix}
b\\
-3-\frac{2}{3}b
\end{pmatrix}=\begin{pmatrix}
a+b\\
-6-\frac{2}{3}(a+b)
\end{pmatrix}
$$
and we see that the addition does not have the form of a vector in $H$ since it is impossible that $-6-\frac{2}{3}(a+b)=-3-\frac{2}{3}(a+b)$
Note that in your answer you have used two vectors that are not vectors of $H$ (so the answer is wrong), but you can chose any two vectors of $H$ as e.g. $(-3,0)^T$ and $(0,-2)^T$ .
Now you can do the same for the question 3) testing if
$$
k\begin{pmatrix}
a\\
-3-\frac{2}{3}a
\end{pmatrix}
$$
has the form of a vector in $H$.  Can You complete ?
