is-it-possible-that-all-solutions-of-a-homogeneous-system-of-ten-linear-equations-in-twelve-variable

Name: is it possible that all solutions of a homogeneous system of ten linear equations in twelve variable
Uploaded: 2019-08-10
Duration: 3 min 15 s
Channel: Numerade
Description: Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

Question

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

Runpeng Li · Accepted 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.

Tommy Erdos · 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.

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.

Charles Carter · Answer

Okay, So this is just to briefly explain if a consistent system of linear equations, if it&#x27;s possible to have exactly two solutions. So again, the key word here is exactly so for two. Linear, uh, we&#x27;re talking about straight lines, especially in the Euclidean space. Um, we have that option for two straight lines. We definitely know that we&#x27;re the Intersect. At one point, we have the option where they&#x27;re parallel as so this one. This one was one. Hey, Parallel has zero solutions. So exactly two solutions. The lines about the cross twice and on Lee twice. The only other option we have is where both lines are describing the same equation. But in that case, we have an infinite number of solutions. So as long as we&#x27;re only talking about linear equations, there is not a possibility to have just two solutions. Um, it might be confusing to think about like could it have two solutions? Yes, but it would also have infinite solutions. So the key word that we want to talk about is this exactly. So there is not just two solutions when we&#x27;re talking about any one of these options. Thank you very much.

Christopher Stanley · Answer

get on this. Probably want to know if it&#x27;s possible for a system of linear equations. Toe have two solutions, exactly two solutions. But that&#x27;s not possible, because our only options are if we draw this straight line and drawn another straight line, those only intersected one point, so that would just be one. Or we could have parallel lines and those have zero solutions. We&#x27;re you&#x27;re gonna be the same way. There&#x27;s a lie on top of each other, and I was supposed to lie on top itself in the same way. So just lies on top of itself, so that would have infinitely many solutions. But in none of these do we have exactly two solutions, and so it means it&#x27;s not possible.

Problem

A homogeneous system of twelve linear equations i…

Problem 22 Medium Difficulty

Is it possible that all solutions of a homogeneous system of ten linear equations in twelve variables are multiples of one fixed nonzero solution? Discuss.

Answer

it is not possible.

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

Linear Algebra and Its Applications

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it is not possible.

Linear Algebra and Its Applications

Kayleah T.

Caleb E.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Kayleah T.

Caleb E.

Samuel H.

Joseph L.

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

Linear Algebra and Its Applications