in-exercises-3136-respond-as-comprehensively-as-possible-and-justify-your-answer-if-q-is-a-4-times-4

Question

In Exercises 31–36, respond as comprehensively as possible, and justify your answer.
If $Q$ is a $4 	imes 4$ matrix and $\mathrm{Col} Q=\mathbb{R}^{4},$ what can you say about solutions of equations of the form $Q \mathbf{x}=\mathbf{b}$ for $\mathbf{b}$ in $\mathbb{R}^{4} ?$

Melvin Adkins · Accepted Answer

So we have that Q is a Square four by four matrix and that the column space of Q is equal to form. But there are only four columns of Q. So this means that the columns of Q are linearly independent. Therefore, a Q is in vertebral by the in vertebral matrix there. Um, moreover, Q X equals B has the unique solution X equals the inverse of Q times. Be, um, for each B and R four, and that&#x27;s by three and five of section 2.2, and that completes the problem.

Gideon Idumah · Answer

So for this question, say&#x27;s the no space off B is the zero subspace That is, of course, to zero. Then this implies that speedy multiply by extra quarter zero as only the trivial solution as only a trivial solution. What I mean by this is dots only X equal to zero. That&#x27;s what I mean by tree Vo con Mick P. Times X equals zero. No, says p squared. Then we can use the invincible metrics Xyrem. Not since section Superman 30 and we have That&#x27;s hey is invincible. And what this implies is that P eggs equal Toby as unique solution for every be in our five.

Gideon Idumah · Answer

answered this question. We&#x27;re gonna rely on serum aids in section 1.7. I&#x27;m gonna waste it sits here. It states dubs it said&#x27;s contains, um uh, vet SOS Done. There are entries in each vessel, then decides is linearly dependence. That is, in other words, what moment is that? If we have this said&#x27;s V one up VP in our end, this sets is dependence that is literally dependents. If that is, if you off if there are more entries. If if its contents more vector than the entries, that is. If P is big garden and no, let&#x27;s go back on solve this question. So we have that sense. The columns off, eh? Form busies for our M. This would imply that&#x27;s because for them to form a business, they are linearly independent. It&#x27;s implies that the columns off, eh? Our linearly independence. I did not independent since then, bite serum aids that I stood it up move are both. It implies that&#x27;s did can not of more columns done rose rights because the number off columns is end on doctors like D. Number of letters in day on the number off entries is a number of rules. So that implies dots dot sees. Never write this in red in Green Carlo and us to be less than a week or so. And now let&#x27;s continue no seas the columns off, eh? Form a busies for our M This implies dots The columns off eh spawns our am because the definition of businesses that it&#x27;s most Belinelli independent on its most spon the subspace So we have that the clumps off a span our am now This would imply that a most as a pie votes being it&#x27;s rule now for its for a safari magics to AFP ive what&#x27;s in its rules. This most implied that the number off ruse can not be no done The number of columns rights the number off columns that&#x27;s is what this implies is dots m must be less than her course to end. So now we have in the impinge one we have and is less than a ghost tow am on. Now we have m is less than a ghost to end This most implied that and must be equal to m on. What this implies is that a most be a square matrix must be squared

Michael Jacobsen · Answer

in this video, we start out with the four by three matrix, and we&#x27;re going to consider the equation. X equals B. Let&#x27;s start off by making an assumption. Let&#x27;s suppose that the Matrix Equation A X equals B has a unique solution for some vector be in our for So let&#x27;s next. Consider what that would specifically mean for this matrix. Well, it would imply that in the equation, X equals B If we have a unique solution, there are no free variables, so start there. So that&#x27;s last line implies that there are no free variables. How do we know this? Well, if there was a single free variable, we would have infinitely many solutions, not a unique one. So next, knowing that there are no free variables that implies the following. Ah, free variable will always come from a non pivot column of The Matrix, say, so we know that A has a pivot in all Collins, and there&#x27;s three com specifically. So let&#x27;s say all three columns squeezing the three in there. Sophie has a pivot in all three columns, but it is of size four by three. This is our next immediate conclusion. I will say that a must be equal to or rather not equal but row equivalents to a four by three matrix. That would look like this so we can have 100 now. Calm One is a pivot column for row to go 010 Now we have to pivot columns, but we require three. So the third row will be 001 and we&#x27;re looking at the reduced row echelon form, so their final row will be all zeros. So given just this minimal set of information, we now know that the role reduced echelon form of a takes on this form.

Gideon Idumah · Answer

So for this question, since the no space off our it&#x27;s not the zero subspace this implies darts our eggs equal to zero gifts in nontrivial solution. And what I mean by a nontrivial solution is that that sees there is so it is an east. There is there is a ex not 1/4 0 such guts. Our X equal to zero No, since are is square, then by D in vegetable metrics serum. This is found in section Super Intern. In your textbook, it&#x27;s implies that are is not in Vaasa ble on. This would imply that the columns off our ah linearly dependence. Because of that, if it is not invincible, then the columns off our linearly dependence. Now the conclusion is, that is the column space off our is a subspace off our six. But since the columns off our legal independence boats, this would imply that column space off. Our cannot be equal to our six

Matthew Allcock · Answer

in this video to go through the answer to question number 25 from chapter 9.5 so as to find the general solution Thio this set off different questions. Okay, so that&#x27;s right. It first in matrix for So if we let the vector X have components X y zed than X ex prime is of equal thio some matrix times by X vector X That is so just fill in here all the components from this of different equations. Okay, so now we sold this unusual way by finding first the Eiken values. And then there are spending corresponding Aiken vectors from that construct the general solution. It&#x27;s it&#x27;s fine. I can values when it&#x27;s finally determine off that matrix minus ah, time by the identity matrix. So that&#x27;s gonna be one minus, huh? Times on the neck. A bonus too. Next. Good. Minus 11 Minus are one four minus four five minus R. Okay, so expanding along the top. We&#x27;ve got one minus R because by the tip of the bomb, right, two by two matrix. Well, I guess we can just go straight for that. That&#x27;s gonna be minus ah times by five minutes. are AA minus minus four, which is pushed for okay than minus two times by descendant of the Matrix won one for five minus R. So it&#x27;s gonna be five minus, uh, minus four and then minus one times by the December off the matrix one minus. R for minus R. That&#x27;s gonna be minus. Mmm. Minus four times my sources close for okay, that&#x27;s that&#x27;s gonna be equal to one minus, uh, times by. Ah, Squares minus five. Uh, plus four minus ah. Minus two times minus size plus thio. And then five minus four is 1000 minus two is minus two. Man minus. Ah, one times minus four. It&#x27;s plus before minus for Ah. Okay, so this is gonna be a cool minus, huh? Go ask. Wet, minus five, uh, clothes for. And then what we&#x27;re gonna have here. So we&#x27;ve got to aa minus fora gives us minus thio at minus two. Goes boys plus two. Okay, so then we could take out one minus side is a common factor. And then we got our squared minus five. Uh, plus, for course, one sorry plus folk was, too. Maybe which I say I focus too. Is six. Okay, so then sure. What? The one minus r. And then we can just fact arise this quadratic. So we got our, uh, see, that&#x27;s gonna be our minus two. Uh, ballistic trips cost three because, uh okay, wait. I&#x27;m in a second. That&#x27;s gonna be I want us to all my industry because a minus two minus three at minus five in the times to make six. So if we then set our equals zero, then we&#x27;re gonna have my own values. I was equal to one, two and three. Yes, and I was known. It&#x27;s finally I connected, associating that values for the 1st 1 r is equal to. So we need to find the matrix. Ah, a minus. One times that enter matrix. So that&#x27;s gonna give us zero to minus one one, minus 11 for minus four and five miles. Once for times buy you one. It was zero. So the first site that second under third row our equipment, um so we can forget about the third World from the first road. We can see that if, uh, this relates the 2nd 3rd components off that I come back that you want, so If the second opponent is equal to one that certain burn, it must be able to. Okay, so then using the second row was this tellers. It tells us that the first component is equal to the second Carbone of minus that the promoter. So that&#x27;s one minus two, which is minus what? That&#x27;s our first Eigen sector. So a sec monk we found with the second item values are is equal to two. That kid&#x27;s a matrix of minus one to minus one one minus 21 for months, for three times by you too equal, Sarah. Okay, so this one&#x27;s a little bit more tricky. So to find the opponents off you two, let&#x27;s do some role reduction. So this matrix can brokerages to Okay, let&#x27;s see. We&#x27;ll weaken. Yeah, the second. Well, weaken just set. Equals is there is. There is zero. Because we could just at the first words second word get, the first rule is and then what could we do? We can have the two times the first rose in the third row, Binoculars to zero Ah, one And then we could really use again. So what will it be this time? So we can add one of the third row to the first room. So that gives us one two zero. And of course, 000 No, 201 Okay, so now we just read off what you two is gonna pay. So you two is gonna bay? Well, if the Okay, let&#x27;s say the second opponent is one. Then the first row will tell us that the first bones really minus two. Uh huh. On the third opponents think to tell us that yet Third West motels that the opponent is going to be minus two times Manitou. It is for okay, then finally, I&#x27;m value our secrets of three. That&#x27;s when he was a matrix minus 22 minus one. Well, monastery one for minus four. Two. Close by a vector, You three. Okay, so again, we&#x27;re gonna have to do a big production here. So this is gonna it&#x27;s gonna blow over drinks too. Okay, what we do here. So the third row is just equivalent to the first road, so I just set out to be zeros. It&#x27;s not going. How was it all? Let&#x27;s keep this second row as it is, then let&#x27;s abs Two of the second room, but less at one of the second round of the first. Okay, let&#x27;s can give us minus one minus one zero than Burma. Juice again on Dhe. Okay, this time. But one of the first road to the second room. I was gonna get rid of that guy, huh? Okay, so now we should be able to read off what Prince of you three are. It may be so if we set the second opponents we want, Then the second row tells us that the 3rd 1 is gonna be four, then also, the first components gonna be minus what the 2nd 1 is. What one? Okay. All right. So now we&#x27;ve got our three agonize on our three Aiken vectors. So the vector x y zed Gemma solution is gonna be It&#x27;s gonna be a constant multiplied by Aedes the first Aiken value. It was one Times team. It&#x27;s nice by the first, Like vector, which waas minus one one two plus a second constant C two times by E to the to T. That&#x27;s our second. I&#x27;m value sounds about second Aiken Vector, which is minus two one four plus C three with us third constant times by eat the three. T three being our third Aiken. Value tells about third I&#x27;m backed up, which is minus 11 Cool, that&#x27;s a generous lesion.

Problem

In Exercises 31–36, respond as comprehensively as…

Problem 33 Easy Difficulty

In Exercises 31–36, respond as comprehensively as possible, and justify your answer.
If $Q$ is a $4 \times 4$ matrix and $\mathrm{Col} Q=\mathbb{R}^{4},$ what can you say about solutions of equations of the form $Q \mathbf{x}=\mathbf{b}$ for $\mathbf{b}$ in $\mathbb{R}^{4} ?$

Answer

If $C o l Q=R^{4},$ then the columns of $Q \operatorname{span} R^{4} .$ since $Q$ is square, the IMT shows that $Q$ is invertible and the equation $Q x=b$ has a solution for each $\mathrm{b}$ in $R^{4} .$ Also, each solution is unique, by Theorem 5 in Section $2.2 .$

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

Linear Algebra and Its Applications

Grace H.

Heather Z.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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If $C o l Q=R^{4},$ then the columns of $Q \operatorname{span} R^{4} .$ since $Q$ is square, the IMT shows that $Q$ is invertible and the equation $Q x=b$ has a solution for each $\mathrm{b}$ in $R^{4} .$ Also, each solution is unique, by Theorem 5 in Section $2.2 .$

Linear Algebra and Its Applications

Grace H.

Heather Z.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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

Grace H.

Heather Z.

Alayna H.

Kayleah T.

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

Linear Algebra and Its Applications