Regarding notation of subsets Say there is some subset in vector space $ℝ^4$ denoted by the following:
$$\left\{(x_1, x_2, x_3, x_4) ∈ ℝ^4 : x_1 + 5x_2 - x_3 + x_4 = 0\right\}$$
In layman's terms, what does this statement mean? I understand that there are four elements of the subset and that they are all within the subspace of vector space $ℝ^4$, but the latter portion of the statement is not notation I am familiar with.
 A: There are a few points to mention here. Firstly, the expression in the curly brackets is not a statement per se, but rather an expression describing what the subset/subspace is. In layman's terms, the expression denotes the set of all points in $\mathbb R^4$ (to begin with, think of $\mathbb R^4$ as four-dimensional space) such that the first component plus five times the second component minus the third component plus the fourth component is zero. There are infinitely many such points. For example, the point $(4, -1, 1, 2)$ is one example, and you should try to think of some more. So when you say there are four elements of the subset, there are actually infinitely many! Further, the subset is three-dimensional, since
$$\left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right] = \left[ \begin{array}{c} -5x_2+x_3-x_4 \\ x_2 \\ x_3 \\ x_4 \end{array} \right] = x_2\left[ \begin{array}{c} -5 \\ 1 \\ 0 \\ 0 \end{array} \right] + x_3\left[ \begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array} \right]+x_4\left[ \begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array} \right]$$
Lastly, be careful to keep in mind the distinction between subset and subspace. It turns out that the subset in question is also a subspace of $\mathbb R^4$.
A: Quite often the colon in the middle of this expression can be read "such that".
The portion to the left tells you what kind of elements the set has - here they are ordered quadruples of real numbers.
The portion to the right expresses conditions - here that a specific linear combination of the co-ordinates is equal to zero.
We might read "the set of ordered quadruples of real numbers such that this linear combination of components is equal to zero" or (more naturally here) "the set of ordered quadruples of real numbers with the property that this linear combination of components is equal to zero".
