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I am trying to find a basis for the region $$-1 \leq x + y \leq 1$$

I am thinking of letting $X = \{ (1,-1)t+(0,1):t \in \mathbb{R} \}$ and $Y = \{(1,1)t+(0,-1):t\in \mathbb{R} \}$ and does $X + Y$ represent the region?

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Can you explain what you mean by basis of a region if you can. I am not familiar with the concept. –  dineshdileep Mar 25 '13 at 5:49
    
If possible I want to find a set of linearly independent vectors that span the region. Which now that i think about it, it probably is the sum of the two spaces... –  sidht Mar 25 '13 at 5:55
    
@sizz, usually we only talk about basis with regard to a vector space. So your concept sounds strange. And for this two-dimensional vector space, you just need to take any two linearly independent vectors, and that will generate the whole vector space, and hence your particular region. Does this answer your question? –  Easy Mar 25 '13 at 5:59
    
No, but I actually want a basis for just that particular region and nothing outside it. The region should describe a triangle –  sidht Mar 25 '13 at 6:09
    
According to your description, you will only get a band bounded by two parallel lines. And if you want to represent this region, you could simply take one vector that are parallel to the boundary, say $(0,1)+(t,-t)$, which is your $X$, and take another linearly independent vectors whichever that can reach the other boundary, say $(0,1+t)$, but you SHOULD add on the condition that $t\in[0,2]$, not any real numbers! –  Easy Mar 25 '13 at 6:31

1 Answer 1

up vote 3 down vote accepted

Your Region is not a triangle, here is a plot of this region enter image description here

it is not bounded, but that shall give the idea.

This one was made with Mathematica the code is

RegionPlot[-1<= x+y <=1,{x,-10,10},{y,-10,10}] 

which can be used in Wolframalpha too. It is not possible to find a basis in the sense of a basis of a vectorspace, as it won't be closed under addition nor under multiplication with a scalar. Still you can represent it as $$M=\left\{ x\in \mathbb{R}^2 : x= \lambda\cdot \begin{pmatrix} -1 \\ 1 \end{pmatrix} + \gamma \cdot \begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} \text{ with } \gamma \in [-0.5,0.5], \lambda \in \mathbb{R} \right\}$$ So you can describe every point in the region unique with the parameters $\gamma$ and $\lambda$, but this is not a basis in the sense of a vector space.

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How did you deduce that $\gamma$ needs to be within $]-1,1[$? –  sidht Mar 25 '13 at 19:35
    
@sizz oh sry forgot to answer, the thing is that $(-1,1)$ always gives you a zero in (x+y) and oh you are right it needs to be [0.5,0.5] i will change it –  Dominic Michaelis Mar 26 '13 at 6:11

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