Linear Algebra: Subspaces In my text it's not very clear as to what the procedure is for determining when a vector is a subspace in say the subset $R^3$ in this example:
Consider the vector of the following form
$\begin{bmatrix}
       a           \\
       b \\
       1
     \end{bmatrix}
$ is this a subspace of the subset $R^3$?
My best understanding is that I do something like this:
$u = \begin{bmatrix}
       a_1           \\
       b_1 \\
       1
     \end{bmatrix}
$ and 
$v=\begin{bmatrix}
       a_2           \\
       b_2 \\
       1
     \end{bmatrix}
$
(Quick Question: $\oplus$ and $\odot$ what does it actually mean? Since I'm currently just using it based on the other examples without an understanding for what it really means)
Now $u\oplus v =\begin{bmatrix}
       a_1 + a_2           \\
       b_1 + b_2 \\
       2
     \end{bmatrix}
$ Now since there is no way to use  this to express $c \odot \begin{bmatrix}
       a_1           \\
       b_1 \\
       1
     \end{bmatrix} $ it's not a subspace then.
 A: First, regarding the terminology: One wouldn't usually speak of a vector being a subspace. A more usual formulation of your question would be: Do the vectors of the form $$\begin{bmatrix}
       a           \\
       b \\
       1
     \end{bmatrix}$$ with $a,b\in\mathbb R$ form a subspace of $\mathbb R^3$?
Your negative answer to this question is correct. Your argument is also correct but a bit of a detour, since you're using the fact that in a subspace any vector can be expressed as the sum of two vectors in the subspace. That's true, but it's easier to argue directly from the definition. You could use either $u+v$ or $cv$ with $c\in\mathbb R\setminus\{1\}$ to show that this is not a subspace, but you shouldn't compare these two cases with each other, but rather argue that in neither case the result is of the given form (since it doesn't have a $1$ in the third component).
Regarding your notation for the operations, I've never seen that before; I suspect it's specific to the text you're using. I presume that they're using these symbols to distinguish vector addition and scalar multiplication of vectors from addition and multiplication of real numbers.
A: Another way to see that it's not a subspace is that a subspace of ${\bf R}^3$ must contain the zero-vector, $(0,0,0)$, and your subset does not. 
