in-exercises-5-8-find-the-minimal-representation-of-the-polytope-defined-by-the-inequalities-a-mat-3

Question

In Exercises $5-8,$ find the minimal representation of the polytope defined by the inequalities $A \mathbf{x} \leq \mathbf{b}$ and $\mathbf{x} \geq \mathbf{0}$ . 
$$
A=\left[\begin{array}{ll}{1} & {3} \ {1} & {1} \ {4} & {1}\end{array}ight], \mathbf{b}=\left[\begin{array}{l}{18} \ {10} \ {28}\end{array}ight]
$$

Chris Trentman · Accepted Answer

were given The Matrix and a column vector were asked to find the middle representation of Polito, represented by the Inequalities X is less than or equal to be and X is greater than or equal to zero where A and B are the matrix and column that they&#x27;re respectively. So The Matrix A is 131141 in the column. Vector B is 18 10 28. So we obtained three inequalities. X one plus three x two is less than or equal to 18. I&#x27;ll call this inequality a inequality be, which says that X one plus x two is less than or equal to 10 and inequality See which says that four x one plus sex to is less than or equal to 28. Now we see the line A goes from the 0.0 six to the point 18 0 since X greater than equal to zero tells us that Polly Tope is going to be in the first quadrant. We have that mind be goes from 0 10 to the 0.10 0 and finally line See goes from the point c 0 28 to the point 70 now again because X is greater than a quarter zero. We have that 00 is going to be a vertex of the simplex or polito Pennine. Now the X one intercepts these air one x two is zero these air going to be 18 10 and seven. So it follows that 70 is a vertex. Yeah. Now the X two intercepts, which is one X one, is equal to zero. These are 6, 10 and 28. So it follows that 06 is a vertex of the poly tube. Now let&#x27;s find the intersection of the lines. We see that lines A and B intersect well. We can simply subtract B from A to get two x two equals eight or X two equals four and then from that x one equals six now So we have 64 is a point of intersection and likewise we see that 64 is also a point that lies in line. See, So it follows that all three lines intersect at the 0.6 four so it follows its 64 is a vertex and putting this all together The minimal representation is 00 70 06 and six four

Chris Trentman · Answer

We&#x27;re giving Matrix A and A column Vector B and were asked to find the minimal representation of the polito to find pay inequalities. A X is less than or equal to be and X is greater than or equal to zero. In this case, were given the Matrix A equals 1231 and be is the vector 10 15. So you see the two inequalities from this matrix are X one plus two. X two is less than we go to 10 and we also have that three x one plus x two. It&#x27;s less than or equal to 15. This is what a X is less than or equal to be means. Yeah, we can represent thes using letters, so this will be equality. A inequality be. We see it the line for inequality A, which is X one plus two x two equals 10. This is going to go from the point 05 to the point 10 0. This is because X and Y coordinates must be greater than zero. Since we have that X is greater than or equal to zero. Must be in the first quadrants, in other words, and we see the likewise line be goes from the point zero 15 to the point 50 Now we have that also because X is greater than equal to zero. That one Vertex is 00 We see that the X one intercepts thes air one next two is equal to zero are 10 and five and so it follows that 50 is a vertex. Now the x two intercepts this is one x one equals zero are five and 15. So it follows that 05 is also a protects. Now the two lines A and B When do they intersect, the lions intersect. This is a simple system to solve and we see just by looking at it, we can almost solve it, multiplying the lower equation by two and subtracting from the top equation. We get negative five x one equals 20. So that gives us X one equals four. And then that means X two is going to be three say intersect at 43 So it follows that for three is also a vertex and therefore the minimum representation is going to be all these Vergis ease. So 00 05 50 and 43

Chris Trentman · Answer

were given a matrix and a column vector were asked to find the minimal representation of the polito to find by the inequalities a X less than equal to be and X greater than or equal to zero. Where is the matrix and B is the column Vector Matrix A Is 2341 column. Vector B is 18 16 and we require that a X is less than or equal to be. And that X is greater than or equal to zero Second requirement really tells us, were restricted to the first quadrant, so mhm we see that the two inequalities are two x one plus three. X two is less than or equal to 18. Call us inequality A and their second inequality be is four x one plus x two is less than or equal to 16. This is what X let&#x27;s drink would be means. Now we see that the line for inequality A. This goes from the point zero six to the point 90 We see that line be goes from 0 16 to the point 40 now, because X is greater than or equal to zero. We have to include the possibility the X zero and therefore we had that the origin is a vertex. Now the X one intercepts the user. One x two is equal to zero. These are nine and four. So it follows that 40 is a Vertex. Likewise, the X two intercepts which are when x one is equal to zero. These are six and 16 so it follows that 06 is also a vertex. Now let&#x27;s find where the two lines intersect. We can do this quickly. Looking at her two lines. Take the equation A and then we&#x27;re going thio multiply it by two and subtracted from equation Be so we get one minus six and negative five x two is equal to 16 and then minus 36 is negative 20 So negative five x two equals negative 20 or X two equals four And therefore, after arranging, we get the X one is three. So I see it. The two lines intersect at the 0.34 so it follows that 34 is also a vertex and therefore the minimal representation for this set is vector 00 Vector 40 Vector 06 and the Vector 34

Chris Trentman · Answer

were given a matrix and a column vector were asked to find the minimal representation of politics defined by the inequalities. A X is less than or equal Toby and X is greater than or equal to zero. Where is the Matrix and B? Is the column back now? The Matrix A is 211112 and the column Vector B is 867 So we obtained the three systems, or three inequalities. Two X one plus x two is less than or equal to eight. Shall call inequality a inequality be, which says that X one plus x two is less than or equal to six and equality. See which says that X one plus two x two is less than or equal to seven. Now we see that because we&#x27;re restricted to the first quadrant by X greater than or equal to zero, the line for inequality A goes from point 08 to the 0.40 Likewise, we have that line be goes from the point 60 206 or the other way around. However, you wanna look at it and finally line See goes from the 0.0 seven halves, which, if you want, can rate is 3.5 to the point 70 now because X is greater than or equal to zero. Well is that X equals zero is possibility And so the origin is a vertex of the set. Now we see that the X one intercepts these air The points on the X one axis this is one x two is equal to zero. So these are four, six and seven. So it follows that 40 smallest of thes is a vertex. Likewise, the X two intercepts thes are eight six and 3.5, so it follows that 3.5 is also a vertex. Now let&#x27;s find the intersection of these lines. This is a fairly simple system to solve. Just looking at it, we see that if we subtract the second check to be from a we get the X one is equal to two and plugging back in. We get the X two is four. So we have two fours a point on lines A and B. We also see that plugging into four. Yeah, one second My mistake. So let&#x27;s find some intersections. We see that a and B, you&#x27;re going to intersect. As we determine before 24 However, we&#x27;re plugging into four to be third equation. See inequality. We see that two plus eight two plus eight his 10 which is greater than seven. So it follows that to four is not a vertex. It&#x27;s not even the set. Consider instead the intersection of lines A and see they intersect at the point. What to do is simply multiply C by two. Subtract it from a so we get one and then minus four. So negative three x two It is going to be equal to eight minus 14 which is negative. Six. So negative three x two equals negative six or x two equals to therefore X one to is to so x one is going to be four. So these intersect at 42 But again, by the same reasoning for two, this doesn&#x27;t satisfy the second inequality. Well, it does actually didn&#x27;t satisfy the third, so I&#x27;m not sure how we got there. Hm. Okay, think about this. A second eight minus 14 is negative. Six and then to four to make it three negative six. So we get to then listed too. It becomes the three okay, minus take should be the Intersect at 32 And in fact, we see that three plus two is less than or equal to six. So it follows that 32 is in fact a Vertex. And therefore, the minimal representation of this Polito is 00 followed by 40 all of by 3.5 and then by 32

Jeyasree R T · Answer

given compound inequality. The given component equality is minus three, less than or equal to three x minus fight less than minus one. We need to isolate X in the middle. So as a first step, let us add fight toe All three parts off this inequality we get minus tree less fi. Yes, no or equal to you. Wait three x minus Fight. Let&#x27;s fight less than minus one less fight. So this is nothing but to listen or equal to Dubai three x Less than four. No. Again to isolate X in the middle, let us multiply that A simple local off to a tree that this three by two to all three parts off this inequality so much playing three by two we get three by toe in tow to does the naughty quarter to three right toe in Dubai, three eggs less than righto and do four. We get three less than or equal to X less than six. So the solution ist set of all real numbers. X. The door rhetoric will do three and less than six in some building tradition. This is the present address. X is the real number such event three Let&#x27;s not equal to X last time. Six and in a double notation, it is the press interest free. Come on. Six. Since the three is included in the interval, Michelle represented, disclosed and here six does not included changeable. So mission the prisoner desist open hence the solution.

Jeyasree R T · Answer

I even called won&#x27;t inequality, but given compound inequality is minus living less than two x minus one that&#x27;s not equal to minus straight. We need to isolate this X and imminent. So as a pastor, let us have one toe all three points off this component inequality. They get minus 11 less, one less than do X minus one. This one. Listen, Arctic will do minus way less one that does nothing but minus 10 less than two weeks. Let&#x27;s not equal to minus sport. Now again, to isolate this ex, let us divide the whole inequality. By two we get minus stand by two Liston two x by two does not equal to minus four by two. When simplifying this, we get minus fight Less than X does not equal to minus two. Hence the solution ist set of all real numbers X There are greater than minus by and less than or equal to minus two. Instead building tradition. This is the present illness existed, you know, such a bed minus by less than X less than or equal to minus two on an internal addition. This is the press interest minus bite. Come on, minus two since here minus five is not included. We shall a percentage in this open interval and hear this minus two is included. So we shall have presented as lost in Tripoli, hence the solution.

Problem

In Exercises $5-8,$ find the minimal representati…

Problem 7 Easy Difficulty

Answer

$\left\{\left[\begin{array}{l}{0} \\ {0}\end{array}\right],\left[\begin{array}{l}{7} \\ {0}\end{array}\right],\left[\begin{array}{l}{6} \\ {4}\end{array}\right],\left[\begin{array}{l}{0} \\ {6}\end{array}\right]\right\}$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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$\left\{\left[\begin{array}{l}{0} \\ {0}\end{array}\right],\left[\begin{array}{l}{7} \\ {0}\end{array}\right],\left[\begin{array}{l}{6} \\ {4}\end{array}\right],\left[\begin{array}{l}{0} \\ {6}\end{array}\right]\right\}$

Linear Algebra and Its Applications

Catherine R.

Kristen K.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Kristen K.

Samuel H.

Joseph L.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications