# So what does x in RR^n mean when discussing homogeneous matrices? I have never seen this notation before and yet it was used in one of my linear algebra lectures. We were talking about homogenous matrices. I was given the following definition and proposition. Definition: The set {xin RR^n|Ax=0} is the null set of A. Proposition: If p is a vector such that Ap=b, then {x in RR^n|Ax=b} = {y+p|y in NS(A)} I am just so perplexed by the sudden introduction of notations and all of the null set. Can someone please explain this in simple terms and explain what the notations are??

I was given the following definition and proposition.
Definition: The set $\left\{x\in {\mathbb{R}}^{n}|Ax=0\right\}$ is the null set of A.
Proposition: If p is a vector such that Ap=b, then $\left\{x\in {\mathbb{R}}^{n}|Ax=b\right\}$ = $\left\{y+p|y\in NS\left(A\right)\right\}$
I am just so perplexed by the sudden introduction of notations and all of the null set.
Can someone please explain this in simple terms and explain what the notations are?
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Jaelyn Levine
I would guess that this is the first time that you have encountered set builder notation. The expression $\left\{x\in {\mathbb{R}}^{n}\mid Ax=0\right\}$ can be expressed in English as "the set of x in ${\mathbb{R}}^{n}$ such that $Ax=0$". Here, ${\mathbb{R}}^{n}$ refers to the set of all column-vectors that contain n real numbers.
To put this another way, the null set of A is the set of solutions to the equation
$\left(\begin{array}{ccc}{a}_{11}& \cdots & {a}_{1n}\\ ⋮& \ddots & ⋮\\ {a}_{m1}& \cdots & {a}_{mn}\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ ⋮\\ {x}_{n}\end{array}\right)=\left(\begin{array}{c}0\\ ⋮\\ 0\end{array}\right),$
where we think of the column of values ${x}_{1},\dots ,{x}_{n}$ as the single vector "x"