a-homogeneous-system-of-twelve-linear-equations-in-eight-unknowns-has-two-fixed-solutions-that-are-n

Name: a homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are n
Uploaded: 2020-07-04
Duration: 4 min 10 s
Channel: Numerade
Description: A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.

Question

A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.

Tommy Erdos · Accepted Answer

this question. They ask us to find out how many equations we need, actually, in order to solve the system of equations. They gave us a system with 12 equations, eight variables, large, right? And so that, Solutions said, has eight variables in it as well. And they tell us that it&#x27;s a homogeneous equation, that which means that it&#x27;s going to equal a zero vector. So that just means 12 zeros, 12 components all equal to 70 12 rows of zeros. No, uh, this will tell us that there&#x27;s within this really only two vectors that matter there to independent fixed solutions is the way that they weren&#x27;t it. Essentially, what they&#x27;re saying is that any answer is just gonna be some kind of linear combination of these two vectors. You can scale them if you want to. You know, essentially, what that&#x27;s telling us is that no matter what we do, we have two vectors that matter. You can also jarred out like this. Now you go back and look at this coefficient matrix. You might remember something called reduced the row echelon form, and that&#x27;s the end up, having the ones with the every single other component in that Colin as a zero. Now, this is a goal of reducing Westerlund form, essentially to isolate one variable on recall that pivot variable. The really. The answer to this question is going to depend on how maney pivot variables we have. If you recall sometimes year, uh, simplifying these and reducing them on the end up with these entire rows of zeros where there&#x27;s no pivot variable, there&#x27;s no useful information, really, what we&#x27;re saying they&#x27;re these just do not matter. And so that&#x27;s why it comes down to the number of pivot variables. It&#x27;s going to tell us how many equations that we need now rank therms helpful year. The rink theory says that the knowledge et of a plus the rank of A is going to equal. And now and, of course, is the dimensionality of our answer. That, for example, if we have you can look at standard form. Here are answers for these kinds of questions is going to be a two dimensional kind of looks like that X and y two dimensions. To use our end right in this situation, we&#x27;re gonna have something like a h maybe, And those are gonna be eight dimensional. Further down in your book, you&#x27;ll see that the nullity of a it really is just referring to the number of free variables, the rank of Ayer&#x27;s air referring to the number of pivot Very bolstered. Remember, that&#x27;s what we&#x27;re interested in. So now, if we could just find the number of free variables will be very close to her. Answer. Now, I call that they gave us to fix solutions. Right on. There&#x27;s these two numbers we could use to scale these These two actually are what are important here. These they&#x27;re called for you. Variables. They&#x27;re called free because you can put in any real number for these. Multiply at times the vector. Um, and you&#x27;re gonna end up with some answers that some combination of these two that is also a solution. And so if we think back to this matrix here you have eight variables. They&#x27;re really only two options. You can never be free or pivots. And that&#x27;s essentially the same thing that the rank three missing saying we&#x27;ve got to free variables how maney pivots variables do we have if we have the total eight variables advancers six. That means are reduced or echelon form is going to look like that. Got six of these pivot variables of the rest of these equations air zeros or don&#x27;t have pivot variables. They don&#x27;t matter. All you need is the six equations.

Runpeng Li · Answer

for a problem problem 22. So where we need to figure out what it is possible for it is possible that also shows up a homogeneous system up a 10 linear equations with 12 variables are multiples off one fixed, none zero solution. So, actually, this is equivalent to say we have a matrix that is 10 by 12 and we have the phone so much in his system, X equals zero. So this is equivalent to say it&#x27;s impossible for a for A to have dimension open all surveys one. Because if we if every every solutions in this system can be expressed as a multiple upper uppers, upper multiple off one&#x27;s fixed and that&#x27;s your solution, that means every solution can be express as this form. So we must have ah, re variable. We don&#x27;t We don&#x27;t know which one it is, but it must. There must exist a free sample so that every variable in this factor will be determined by this variable. So so she can be expressed as a multiple wall boat about non zero, a fixed and then zero social. So Okay, so what can we see to this extent? So, um, we first by wreck fear. Um, I wrecked your, um We know that rank a fe us. You mentioned nothing. All space. Okay, ISS 12. So we suppose suppose you mentioned on the north face is one this one, Then the rank will be nothing. So is it possible for rank to rack up A to B 11? Well, it&#x27;s none. It&#x27;s not possible because the rank of a should be smaller or equal to 10. But Frank okay, should be smaller. Way to 10. So we have a contradiction. So that means it&#x27;s impossible two.

Subham Jyoti Mishra · Answer

So we have known Home was in a circle nine. Ginette questions in Ben Variables. The coffee is in America. Next video, Uh, size nine and 10. It may be this metrics when the question is do you condemn? So, listen, all these nine set of being with January levels uh, his all of this. Thank adjusting. Let&#x27;s have a listen. He&#x27;s always existing for all the values off constant. So that implies a X is gonna be a desert of equals. And so when be changes so listen, always exist certain place for if the and rule. Oh, so obvious answer texts. The federal off, Uh, as if by work element. So there were ranked off metrics air and sea coral number off course. So they got a nine, then. This is nine. Using ranked Huron, we have and minus rank off here, support diamonds And how you know here, here, this m cross and physical. 9. 10. So when is the European diamonds and uh huh No. Yeah, it&#x27;s headquarter. Nine minutes. Return minus nine is a good one. I wondered this Live by that diamonds and of Middle Ages one So the soldiers in Set X has singer, non zero victor and all of the solutions, um, our peoples of director. So we cannot have to non zero citizens off Saturday quiescence, which are north multiples of digital. So there is only D one non zero Ferguson. There cannot be, Yeah, 20 small business which are no amount of people off each other.

Subham Jyoti Mishra · Answer

mhm circle. I huh, Genius. Linear equations in six variables. So the metrics were responding to the proficient off these equations. It would be in my tricks for size 56 No question have been asked. Suppose it&#x27;s all listens off the set off Jenna represents of all more people off one non zero citizen. So this artist in Service X says they&#x27;re p it&#x27;s Is it gonna be Onda so many pics as the one and on zero. So, listen, this implies there are diamonds or not nerves. Space up here is a good one. And by using drink among building and support Oh, diamonds and off Mother apis rank. Uh huh. So here and you gotta fly him in 0 to 6. It was six days ago. Yeah, one plus Frank is the bank people. Hey, Sequel. Five years. Mhm. Yeah, So Mm hmm. Number off non mus eagle. Who&#x27;s been early? Grand formal, which is? He is equal toe fight. And therefore, but bye bye. Gotta lemon, sir, present in a little. So embryo is a It&#x27;s a total of five rolls are there and there are five fibers. So every room Mhm. This is a for a number. Well, by that very much. And this is also for a number Off goes. So since number of rule number five, therefore duh yeah, system off the business. We&#x27;ll have so here, son. Irrespective off the values off constant.

Shakiyla Huggins · Answer

Okay, so let&#x27;s talk about the number of solutions that every linear equation in two variables have. So the answer is infinite solutions. Every linear equation in two variables have influence infinite Excuse me solutions. And also remember that those solutions are ordered payers.

Sarah Lewites · Answer

so there are many different options for what happens when we have a system of three variables. So it could be that we have no intersections if we have planes lying parallel to each other than we will have no solutions. Sometimes the planes could all be like on top of each other, and so we could have infinitely money. Or we could have all three planes into, except at its one point so we could have one solution. So there are three different options, no solutions, infinitely many solutions or just one solution.

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Is it possible for a nonhomogeneous system of sev…

Problem 23 Hard Difficulty

Answer

the set of all solutions be described with less than 12 homogeneous linear equations.

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

Linear Algebra and Its Applications

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Caleb E.

Kristen K.

Samuel H.

Vectors Intro

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the set of all solutions be described with less than 12 homogeneous linear equations.

Linear Algebra and Its Applications

Lily A.

Caleb E.

Kristen K.

Samuel H.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Caleb E.

Kristen K.

Samuel H.

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

Linear Algebra and Its Applications