How to correctly write the set of all vectors of the form $(a,b,c)$, where $a+2b-c=0$

Question

I'm not 100% sure if this is where this should go, but my LA prof wants us to practice writing more mathematically using correct notation. this is one of our homework problems and I wrote down the problem as $$v=\bigl\{\begin{bmatrix} a& b& c \end{bmatrix}^{\mathrm t},\; a,b,c\in \mathbb{R}\mid (c=1)\wedge (a+2b-c=0)\bigr\}.$$

There are several parts to the question and I think I generalized it well using this for a certain part but I'm not too good at notation yet, so if I'm horribly wrong what would be the correct way to write it?

Answer 1

We have $a+2b-c=0$. Fix the parameter $b=\beta$ and the parameter $c=\gamma$ then $a=\gamma-2\beta$. The points $(a,b,c)=(\gamma-2\beta,\beta,\gamma)=\beta(-2,1,0)+\gamma(1,0,1)$. We can describe the subspace by

$$\mathcal{S}=\{\boldsymbol{x}\in \mathbb{R}^3|\boldsymbol{x}=\beta(-2,1,0)+\gamma(1,0,1) \text{ for } \beta, \gamma \in \mathbb{R}\}.$$

Answer 2

Sets are typically denoted with capital letters, it is efficient to use $$V = \{(a,b,c) \in \mathbb{R}^3 \mid c=1,~a+2b=1 \}$$ or, more succinctly, $$V = \{(a,b,1) \in \mathbb{R}^3 \mid a+2b=1 \}.$$