if-mathbfb-neq-mathbf0-can-the-solution-set-of-a-mathbfxmathbfb-be-a-plane-through-the-origin-explai

Question

If $\mathbf{b} 
eq \mathbf{0},$ can the solution set of $A \mathbf{x}=\mathbf{b}$ be a plane through
the origin? Explain.

Runpeng Li · Accepted Answer

All right, so in this problem, we&#x27;re even. Yeah, it&#x27;s not equal to zero. And we are trying to figure out whether the solution off a X equals B can be the plane through the origin. So in this case, um, I&#x27;ll first tell you that the answer is no. Why? Why is it white is no. So let&#x27;s suppose suppose the solution said passes through as it passes through a region which is started zero vector and then we will prove by contradiction. So since the solution said passes through the origin, then zero has to be a solution. Oh, axe. Because he all right now, now, since zero it&#x27;s a solution. Then we put zero but the value here, the bedroom into the system. So that is eight times zero because be and eight times theories is it is exactly zero. So we get e zero. So he&#x27;s a zero factor here. I recall that we were assuming that it is not equal to zero from our from the beginning. So this is a contradiction. So we&#x27;re done. That means it&#x27;s impossible to have this a plane to have the plane that passing through the origin. If we have the condition here, that B is not equal to zero

Ankit Gupta · Answer

well. C is equal to zero in equation, then modelers off. A X plus B is equal toe. See? Then the equation will have only one solution. Hence, yes, it is possible.

Linh Vu · Answer

in this question were recorded in a Jew determined of plan. We need to information the first one with a pond and another one upon X zero y 00 and another information will be the normal. Me, the any coach, u A B and C And then here we know that because if we don&#x27;t plan here and then if we have the no more listen to me, I being c and this no more here will be always bourbon declared to any light on the on the plan here. So they will be any point energy line and then we&#x27;ll condition will be killed. So if we kinda b will be the X, y and Z and the kid will be the X zero y 00 Therefore, we should get the No. This one will be normal. And so and we&#x27;ll be perpendicular. Julia better be killed on the time. And here we go. And only only if that and that would the Viterbi kill must equal to zero. Here I any wiggle to the A, B and C and that would&#x27;ve better big here will be the X ah minus X zero and then y minus y zero then the Manistee zero ico chew zero. We expand this one with again the eight times x minus X zero plus b times Why minus y zero lost. Ah, see, time&#x27;s the minus zero. We equal to zero. And I will explain this when we gotta I asked minus, uh, blood B y bless Sees that. And then my in us uh, it will minus I x zero minus b y zero minus sees that zero when you go to zero everybody the haunting Here you go to the day some constant day that we can get the final B i x plus b y plus is that anywhere you go thio m plus day equal to zero And this will be the question off the plan. Having a pond x zero, uh, having the bland here having a normal because, you know, I, b and C

Sriparna Bhattacharjee · Answer

mhm. And the question they&#x27;re saying if abc are not all zero to show that the equation X plus v bypasses a plus the is equal to zero re presents a plane and abc is a normal vector to the plane. So if is not equal to zero then X plus B by let&#x27;s see that plus D is equal to zero means that E we take common E. And uh within the common done it is X plus D by a Plus B into Y -0 0 plus Seen to Zed zero equal to zero. So this is the scalar equation of the plane through the point. Okay minus debate. A 00 with normal vector A. B. C. Similarly. Mhm. Mhm. If we put the value of B is not equal to zero so Or if she is not equal to zero then the equation of the plane can be rewritten as equation of the plane can be rewritten as E x minus zero plus B, Y plus being two, Y plus D by B plus see into that minus zero is equal to zero. Or if we put the value of C not equal to zero then the equation can be written as a into x minus zero plus being two, Y minus zero plus Scene two. Yeah they plus D by sea Is equal to zero. So therefore this is the equally scalar equation of a plane. True. The point zero -D by B0 or The zero -D by Sea. People with yeah normal vector. Mhm. A B C. Hence these are the possible equations scalar equations of the plane when abc is not equal to zero, and hence with the normal vector, Abc with the point that the pain passes as minus D by a 000 minus D by b zero or 00 minus D by sea. So this is the answer of the question.

Linh Vu · Answer

in this question would conduct a question on the plan. We need your informations that first will be the one on the plan. Excellent. One&#x27;s own zoos in all. And then no more Victor and A B C therefore confined the question. That man with me, I can&#x27;t x minus. Xterra lost the times one minus y zero the state and Z minus zero coaches zero in this question were given the plan we&#x27;re looking for. We&#x27;re going through the three coordinate Onda, uh, Bond, I, Zo, Zo, Zo and then zero bees. Oh, and then we have a Nonda, so they will see. Therefore, I&#x27;m constantly the A, B and C we can find a better I be And they were equal to the Manus. I mean, there are We can also find a B C and then we equal June. There&#x27;s Omanis PC And we tried your cost. I be because could&#x27;ve BC. We should get a normal better and an ever equal Jew the BC And then this would be, uh I see. And this is the last one will be equal to be. And now therefore, we should be able Joe, uh, we can use the first point I hear and then with the normal better here. So we should be able to find a question of a plan So it would be be seeing temps X minus a plus. I see times why plus I see be tense on the ICO ju zero and in with charges signify gundha BC yanked minus a B c. Unless I see why does I be the ICO ju zero hour ghenda bc x plus I see. Why must I be Is that the culture? And we see from here everything by everything by I b c then which again? The phone the X off I plus why I&#x27;m a big us. See? Let&#x27;s see. I must say it could you want?

Robert Fisher · Answer

we&#x27;re asked which planes can take the form X over a plus. Why ever be plus Z oversee is equal to one? Well, clearly A or B or C cannot be zero because then we would be dividing by zero and that would be else Brickley forbid. So the equation that we have is one over a Times X was won over B times. Why was one oversee Time Z is equal to one. If we look at what values of X, Y and Z, we can have, um or I guess, easier. What we can&#x27;t have. We can&#x27;t have an X equals zero y equals zero z equals zero because then this would be 00 plus zero equals one. So these this plane can&#x27;t contain points of the origin first thing, so can&#x27;t contain points in the origin. But any other ex wives and Z&#x27;s are possible. So what? What if I let x b zero and why be zero? Then I would have 00 was one oversee Time Z is equal to one that would tell me that Z would have to equal see likewise if extra zero and 00 then I would have zero plus one over B times. Why one? Because this is also zero. And why? What equal be? And if why were zero n z zero and I would have won over a Times X is equal to one and ex with equal. Hey! And so what we have is any plane I can take this form provided it doesn&#x27;t go through the origin and it crosses the coordinate axes that hey, B and C.

Problem

In Exercises $29-32,$ (a) does the equation $A \m…

Problem 28 Medium Difficulty

If $\mathbf{b} \neq \mathbf{0},$ can the solution set of $A \mathbf{x}=\mathbf{b}$ be a plane through
the origin? Explain.

Answer

No

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

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Linear Algebra and Its Applications