suppose-the-solutions-of-a-homogeneous-system-of-five-linear-equations-in-six-unknowns-are-all-multi

Name: suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multi
Uploaded: 2020-09-19
Duration: 4 min 7 s
Channel: Numerade
Description: Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Question

Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Subham Jyoti Mishra · Accepted 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.

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.

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.

Runpeng Li · Answer

Okay, So in this problem again, we need to transform the statement to a, uh, to a mathematical statement. So first thing, we notice that we have seven. We have seven equations in six unknowns. So that&#x27;s Ah, seven by six matrix. And we I have another homogeneous system. That&#x27;s a act. People speed. Where is not zero? What? And we want a unique solution for some B for some. Hey, so if we want a unique solution for somebody, that means we we do not want any pre variable. Okay, so in this problem again, we need to transform the statement to a to a mathematical statement. So first thing, we notice that we have seven. We have seventh equations in six unknowns. So that&#x27;s Ah, seven by six matrix. And we have because there exists of repairable, then we will have many solutions. So I want no reversible and further that means dimension of the non space. Okay, Is there a and by wrecked the room we have A is for rank six minus 06 Which, which means age pull rank haven&#x27;t homogeneous system. That&#x27;s a act. People speed where is not zero, and we want a unique solution for some B or some. Hey, so if we want a unique solution for somebody, that means we we do not want any pre variable. You can stick it, Terry. So, you know, reduced reduced a row form. They will be like something here in the first roll Second roll, third row. But the last row last row is always zero. All zeros for a last rule and other rose are none. Zero because we have We know that matrix A as seven rules. So to be fore rank again, it must has exists of repairable. Then we will have even many solutions so want no re venerable and further that means dimension of the non space. Okay, is there and by wrecked the room we have A is for rank six minus 06 which which means a spore wreck. It must have like 6 600 or rose in the road reduced will reduce the Matrix Sebi Okay, If we have such a matrix times say time&#x27;s hacks, then we must have a vector because because the last rule he&#x27;s always dear. Oh then no matter which vector we are considering the last entry off Vector B is always zero and something for so, you know, reduced reduced a row form They will be like something here in the first row Second roll, third row. But the last roll last row is always zero. All zeros for a last rule and other rose are none. Zero because we have We know that matrix A as seven rules. So to be fore rank again, it must has or the rest of entries and we don&#x27;t care. But we are taking that you are taking taking the vector b to be such that the last entries is a road. Then we can actually find they It&#x27;s actually possible to find a unique solution for the system. Okay, It must have like 6 600 or rose in the road reduced. We&#x27;ll reduce the Matrix Sebi Okay, If we have such a matrix times say time&#x27;s hacks, then we must have a vector Just because the last rule he&#x27;s always dear Oh then no matter which vector we are considering the last entry off Vector B is always zero on something for so the next problem Next question you to answer is that whether it is possible for such a system to have a unique social for every right hand side. Um, so to answer to cancer does this question first, I have to I will tell you that this is, uh the answer is impossible. Impossible. Why? It&#x27;s because, for all be we take such a B. That means we are the rest of entries and we don&#x27;t care. But we are taking that. Who are taking taking the vector b to be such that the last century zero Then we can actually find a, uh it&#x27;s actually possible to find a unique solution for the system. Okay, so the next problem next question you need to answer is that whether it is possible for such a system to have a unique social for every right hands on because of the question that asks about every right inside. So if we take some special B that the last entry&#x27;s not zero, say it is one. Then we have some entries for arrested, some numbers for other rest entries. Then we have a system. He acts because fee I remember our reduced form. This is equivalent at reconsider reduced form. It&#x27;s p X. It was B. So I, um so to answer to cancer. Does this question first I have to I will tell you that this is, uh the answer is impossible. Impossible. Why? It&#x27;s because for all be we take such a B. That means, uh because of the question that I asked. Right? So that way remember our last roper? The reduced form is all zeros, so you But our our last entry for this B sector B is not his non zero. So it is impossible. So that means can be done can be solved. Actually, it&#x27;s about every ran inside. So if we take some special B that the last entry&#x27;s not zero, say it is one. Then we have some entries for arrested, some numbers for other rest entries. Then we have a system. He acts because fee I remember our reduced form. This is equivalent at reconsider reduced form. A speedy exit was B

Runpeng Li · Answer

Okay, so problem 20. Suppose a nun. Homogeneous. System up six linear equations in eight unknowns. Has a solution with 23 barrels. Is it possible to change some countenance on the questions right sides to make that uses something consistent. Okay, so, um, what does this mean? The problem means which, actually, this problem is equivalent to say that way, have the matrix A That is, since we have eight unknowns and six lena, you questions that still. So this will be a six by eight matrix. And we have a system. A axe equals B where is not zero. And we know Dad, this matrix has to has to free better balls. So that is so That means the dimension of the north face okay is too. So the question and we are trying to answer is that whether it&#x27;s possible to change some accounts and down the quickness right hand side, make a news has come inconsistent. Well, that is the same to say. Um, well, in this case, we have to choose some Any that makes that this system inconsistent. Um, but by by this information, here we have the dimension. The columns phase okay, is you go to the rank. Okay, well, we can apply the rank serum here. In this case, that will be eight minus two. So that&#x27;s six and further we can apply. The, uh uh, You&#x27;ve earned a matrix theory. Um, remember how we how we stayed, the inverter, inverter? Biometrics, dear, um, that is in these cases, we we will have a mapping from axe to the ex. That that is one too. So what does I&#x27;m too mean. Remember the definition about two? Maybe that is even any or Annie Vector and are six. Because this methods from eyes from our eight are AIDS to our six. So are six. Given any victor from our six, there exists. There&#x27;s always exists a vector from our eight such that the axe he cost. So that means for any animal friend and be the victor in our sex. We will always have a solution to this system. So it&#x27;s impossible to make the system inconsistent.

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.

Problem

Suppose a nonhomogeneous system of six linear equ…

Problem 19 Easy Difficulty

Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

Answer

Hence, the systems have a solution for every possible choice of constants on the right sides of
the equations.

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

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Vectors Intro

Vector Basics - Example 1

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Hence, the systems have a solution for every possible choice of constants on the right sides ofthe equations.

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Hence, the systems have a solution for every possible choice of constants on the right sides of
the equations.

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

Linear Algebra and Its Applications