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.