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Calculus and Beyond Homework Help
Finding the dimension of S
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[QUOTE="Ray Vickson, post: 5509261, member: 330118"] Try to think geometrically instead of algebraically. The dimension will either not exist at all, or else will be one of the integers 0, 1, 2 or 3. Is S a 0-dimensional set? (You can use Google to look up what that means.) Is S 1-dimemsional? (What does a bounded 1-dimensional set look like?) Is S 2-dimensional? What does a bounded 2-dimensionl set look like? Does your S look like that? When you want to approach such problems algebraically, it is more involved and complex than you might think at first. The first step is to eliminate all "algebraic" inequalities, leaving only simple bounds like x >= 1, etc. We do this by introducing so-called slack or surplus variables, one for each inequality. Thus. we re-write ##-2x+y+z \leq 4## as ##-2x+y+z+s_1 = 4##, where ##s_1 \geq 0## is a slack variable. Similarly, ##x-2y+z \leq 1## becomes ##x-2y+z+s_2 = 1##, where ##s_2 \geq 0## is another slack variable. Finally, we re-write ##2x+2y-z \leq 5## as ##2x+2y-z+s_3 = 5##, where ##s_3 \geq 0## is still another slack variable. So, altogether your system becomes [tex] \begin{array}[rcccc] -2x+y+z&+s_1& & & =4 \\ x-2y+x & &+s_2& &=1 \\ 2x+2y-z & & &+s_3&=5\\ \end{array}\\ x \geq 1, y \geq 2, z \geq 3, s_1, s_2, s_3 \geq 0 [/tex] Your system of equations has 3 equations and 6 variables, and that is the system on which you would start to use matrix methods. [/QUOTE]
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Finding the dimension of S
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