let-mathbfv_1leftbeginarrayr0-0-2endarrayright-mathbfv_2leftbeginarrayr0-3-8endarrayright-mathbfv_3l

Question

Let $\mathbf{v}_{1}=\left[\begin{array}{r}{0} \ {0} \ {-2}\end{array}ight], \mathbf{v}_{2}=\left[\begin{array}{r}{0} \ {-3} \ {8}\end{array}ight], \mathbf{v}_{3}=\left[\begin{array}{r}{4} \ {-1} \ {-5}\end{array}ight]$ $\operatorname{Does}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}ight\}$ span $\mathbb{R}^{3} ?$ Why or why not?

Runpeng Li · Accepted Answer

All right. So for problems. Money too. We need to identify whether they given three vectors are whether they&#x27;re given three vectors. Spend our three. So the key thing here is just too. This is just to identify whether these three vectors already nearing end it. If they&#x27;re if they are in leaner independent, then we&#x27;re that because leader independent three vectors will spend our three automatic. So right here. Um, we have we just the first form of this matrix A that is fanned by these three column vectors. So the first column, Mrs 00 Negative too. And zero negative. Three and eight. You and four negative one too far. Okay, um, now we interchange the first row and the third row so that this matrix will be in the Indonesian form. It will be elected to eight. Next five and zero. Negative. Three negative one and 004 All right. Now, if we consider the wiener system a X equals zero, then from the last rule, we have x three times four is zero. So rx three will be zero. And furthermore, from the second rule, we have a negative x two minus one x three that is also zero. But we already have. X three wishes zero So x two has to be zero and similar T r x one by the first row. I won&#x27;t write down the exact question. You just need to know that x one. It&#x27;s also zero. Then are our solution where the system will be zero Dr. So what does that mean? So that means it&#x27;s Sheldon variable in the vector. Acts will not be dependent on each other because we get because we all if we consider the accent zero we the only solution is the trivial solution. So that means that those those three very goes will not will not be dependent on it around them. So that means spinoff. So that means he, uh, three vectors here we want to be three was been our feet. So where

Michael Jacobsen · Answer

in this example, we are provided with three different vectors. Each one of these vectors is an element of the vector space are for And the question we have here is, is it possible that the span of V one, V two and V three could equal are for, so to check? If this is true, we can first consider the Matrix a formed by placing V one, V two and V three into its corresponding columns. If we expand this out, it would be 10 negative 10 for column one calm to is then zero negative 101 and the last column is 100 and negative one. Now, when we consider spanning are four. We know that this matrix a consisting of the columns V. One v two V three. It will spend our four if and only if there&#x27;s a pivot in every row. We could roll reduced her. Check this, but let&#x27;s look before we leap. Let&#x27;s suppose our Matrix A without knowing. Let&#x27;s just suppose it had the maximum amount amount of pivots will best case there be a pivot here. We&#x27;d have to go down a row and get a pivot here, go down a row again and get a pivot here. But this tells us that it is impossible that a has a pivot in every row at this stage. Since if we had 1/4 pivot, we would require an extra column, since we&#x27;d have to move to the right to get to that pivot as well as down. But now, though, we haven&#x27;t that a does not have a pivot. Every row we can say the following The Spanish are four are not equal. We know that these are not equal now, since an order for the span of these vectors to spend all of our four, we would require a pivot in every row. But we also know it&#x27;s impossible for that to happen, so we know that these are not equal and the solution is complete.

Jonathan Turovsky · Answer

Okay, so this problem were asked to reason out if 3.5 squared equals night. According so, the first thing to do is there a 3.5 out as three plus half, right? I&#x27;m sorry. That not 12. That should be helps. So three plus 1/2 squared Does that equal? I 1/4 so we can multiply. Three plus one have squared using our by for a plus B squared. You know, that gives us a squared. Plus two a B plus B squared. All right. It would be just a substitute. Three for a and 1/2 for B you get three squared plus two times. Three times, 1/2 plus 1/2 squared. And does that give us all right. Fourth, Wolfie simplifying three square. That gives us nine plus two times +36 and six times. 1/2 is three. This sometimes was three plus 1/2 squared, which is 1/4 And does that Equal NYT force. And the answer to that is no, because we have this three over here. So this contract people up because they seem okay. Well does night and 1/4 equal three plus one times three plus 1/2 groups plus 1/2. They seem okay. Well, three times three is nine and 1/2 times What? Half is 1/4. So, yes, night and 1/4 does equal nine plus 1/4. But they&#x27;re they&#x27;re forgetting that you have to, uh, they&#x27;re forgetting about the three here because you still have to multiply three times. 1/2 just 1.5. Now that you would have toe plus times three, which is three. So they&#x27;re forgetting about that. Plus three here, so answers.

A M · Answer

So here we have the coefficient matrix A is given by 211 negative. 110 and 333 So this is our A No, we could use matte lip, for example, For mathematica or maple and usedto I God, I command So, for example, if we&#x27;re using mad, let then we can use I command to compute the argon values of this matrix A And so we will get Lambda one or are one equivalently It&#x27;s a lambda one. Will something lend it to equal something and landed three equal something where these air decimal numbers and they have their own associated Eigen vectors. So you want you you to and you three, we&#x27;re gonna be given as since we&#x27;re in Matt Lab, it&#x27;s gonna be given they&#x27;re going to give us normalized Aiken vectors, which means that the Eiken vectors have norm one each or length one and these air some decimal numbers again. Now, we could use all of this information to put together the fundamental matrix No, to conclude the problem as a three by three matrix, and that&#x27;s gonna be given by point for 056 e to the 3.8794 t as the first entry and so on and all of the other entries are similar decimal entries like this, which can be scene from the solution or answer to the problem on the page on numerator where the problem is listed.

Ankit Gupta · Answer

look in this particular question. We have to just show that a plus B. Okay, times a minus bi is equals toe a squared less be okay. Minus a B minus. Bees were okay, So you can clearly start with the solution part. See if I just started left inside. This is a plus B times a minus, bi. Okay, so this is simply is equals to a square minus B squared. And if I just picked the right inside, this is a square. Okay, you can see this. A, B and B are equal because they both are to buy tournament excited. They will cancel out. So this is simply a spare minus beast right to this to our equal, So we can clearly see therefore, a plus B times a minus B is equals to a square. Bless a B minus, B a. The values are equal in this particular question. Okay. And minus B squared hence. All right,

Ankit Gupta · Answer

here. We have to check that one by 30 given polynomial function or north. So given polynomial function f x it equal toe four x cubed minus five X s plan minus three x less one and we have to check that one by trade, Jiro or not. So first we will find all the potential Esnal the roads off people normal function by rest Nalgae rose to Rome. But this totem says that if function is below normal, functional degree one or higher than one, it&#x27;s coefficient off will know millage and Intissar then B by huge personal Jiro off full normal function where he must be the factor off Once Chantome and you must be the factor off leading coefficient off. Um pull normal function to know we will write. Were you off? They will be Quote defectors off Plus one will be ls minus one And the tools you will be Bless Minus one last minus two and plus minus Poor No, we list here All the potential Esnal Gee rose. It will be well to be by two plus minus one last minus one bite too. Last minus one by four. So from here we can see that one by trees nor part off this list off all the potential personal Gerald. So we can say that one by three is north zero given full normal function.

Problem

In Exercises 23 and $24,$ mark each statement Tru…

Problem 22 Medium Difficulty

Answer

since all three rows have pivot positions, then $\left\{v_{1}, v_{2}, v_{3}\right\}$ does span $\mathbb{R}^{3}$
Check theorem 4

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Heather Z.

Caleb E.

Maria G.

Kristen K.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 1

Video Transcript

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

Linear Algebra and Its Applications

Heather Z.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

since all three rows have pivot positions, then $\left\{v_{1}, v_{2}, v_{3}\right\}$ does span $\mathbb{R}^{3}$Check theorem 4

Linear Algebra and Its Applications

Heather Z.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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

Heather Z.

Caleb E.

Maria G.

Kristen K.

since all three rows have pivot positions, then $\left\{v_{1}, v_{2}, v_{3}\right\}$ does span $\mathbb{R}^{3}$
Check theorem 4

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

Linear Algebra and Its Applications