Matematik
F-M elimination problem
Hej
My problem is the following:
Give the complete sets of solutions to P belonging to R^3 for the inequalities:
−1x −1 y − 1z ≤ −1
−5x + y +1 z ≤ 1
−1x + 2y − 1z ≤ 2
1x − 2y + 1z ≤ 1
4x + 4y − 5z ≤ 4.
I tried to solve this with F-M elimination where I first solve for x and then find a interval x belongs to given y and z. Then I solve for y and get an interval y belongs to and then I solve for z. However, here I encounter my problem as z only has a lower bound ?
Also after this I need to give examples of faces for in P with dimension 0,1 and op to dim( P) . I know the basic formula but don' know how I get from the first sub question to the next.
I have made two pages of calculations, I can send you if you don't understand my problem..
Svar #1
08. august 2015 af Soeffi
Jeg får dette i CAS:
Svar #2
09. august 2015 af Eksperimentalfysikeren
I think the best way to handle the problem is to get an impresion of the set as a solid object in space.
Name the inequalities a..e.
In a add 1 to both sides: -x-y-z+1 ≤ 0.
This describes a half space (the dansish term is halvrum) limited by the plane pa: -x-y-z+1=0.
Similar for b..e.
The coefficients of the three variables are the coordinates of a normal vector to the plane:
The normal vector points away from the inner of the pointset because of the less than.
Inspecting the coefficients it is found that nd=-nc. This means that pd is parallel to pc.
The point (0,0,0) is a member of the half space d and of c. When the two normal vectors point in oposit directions, this means that the point set we seek is part of the slice between pc and pd.
Three planes remain. Their normal vectors are neither parallel to each other nor parallel to pc. Therfore they either a more or less triangular piece out of our slice.
The next will be to find the point T where the three planes pa, pb and pe intersect. If T is inside the slice the pointet is a pyramid with T as the top and pc or pd as bottom. Otherwise the set may be empty or it may be a pyramidstub (danish term: pyramidestub). Find the three corners where pa, pb and pe intersect pc and check if they are members of the set. I they are find the similar corners in pd.
I have given the danish terms in two cases. The reason is that I do not know the correct terms in english. I have guest and then given the daninsh terms.
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