in-exercises-9-and-10-find-the-change-of-coordinates-matrix-from-mathcalb-to-the-standard-basis-in-2

Question

In Exercises 9 and $10,$ find the change-of-coordinates matrix from $\mathcal{B}$ to the standard basis in $\mathbb{R}^{n} .$
$$
\mathcal{B}=\left\{\left[\begin{array}{r}{3} \ {-1} \ {4}\end{array}ight],\left[\begin{array}{r}{2} \ {0} \ {-5}\end{array}ight],\left[\begin{array}{r}{8} \ {-2} \ {7}\end{array}ight]ight\}
$$

Doruk Isik · Accepted Answer

in his stroller were given a basis B and B No d lectures that that face is so we have the one we have be too and we have B feet and we&#x27;re us to find the change of ordinance matrix. So we know that in a vector, X is equal to PB times the coordinate vector and this PV is the change of ordinance. Man picks on PB is actually the augmented mentions on by the factors that forms basis beat So pv will be b one b two b three and it is be negative. 1420 negative 58 negative to seven

Arsalan Afkhami · Answer

we need to find the change of basis metrics from B to C. So first we need to equates a fair specter of basis. Be the cheese to negative by zero on. Find the victor that has three components C one c two C three that converts the or the basis see to be so Here we have C one supplied by fair Spektr on one plus c two 20 one plus c three 013 So here your cheeks. Why, I said. And then X is a call to see one by, except the first victor see to my way the first x of the second victor and similarly for the third victor. So we have a system of equations. Three equations see one thus to see to close to two negative C one plus C three, because to negative fine. I see one plus C two plus three seats. Three is close to zero. Here, you can just use one of thes a different color. So you have C three close to C one plus C minus five and then replace C three. Come over here, or we can also use because in elimination and documented metrics to solve it. It&#x27;s ah, just a Z self aversion to solve. So we&#x27;ll find. See one close to four C to close to 91 c three because to unit one on the first Pictoris for a 1 81 So this was the first factor you need to, right? The other one. Jeez, the second Muktar reaches 305 and same s. We did see one by one and round one plus C two Choose your one C three video 13 So should not leave you right. This isn&#x27;t questions for the second victor se see Juan Plus to see to close to three, then the for why he writes native see one plus c three because +20 on lastly, see one plus c two plus three c three these because to fight here we can use C three because to see one So it&#x27;s very simple. You just replace, uh here and then we have system of questions off two equations so it&#x27;s easy to solve on you&#x27;ll find C one c two c three to be one one I want So the second victor would be what is wrong now we need to write the third victor on Equated to the other basis. See? So we have h needed to maybe nine. It goes to I see one bye bye. One the good one plus C two by 20 born plus c three 013 And we write this stuff equations Your I see one plus to see to because to age And if you want. Plus, C three is, of course, a negative, too. See one plus C two plus freeze C three It&#x27;s close Winning nine and yeah, smart before C three Coster Negative too. Plus you wan on, we can replace it here and we will have this itself. A question to a question So four c one c two C three to recall so they have to by on and maybe four. So the fares victor will be native to five and negative four. So these vectors we have weakened three vectors these to about here each are one Colin off the metrics were looking for. So the final, um, answer which is the change of basis metrics from B to c. R. A. T you b two c is a call to on three columns that we had for native want born one on one and then to by negative for and this is the right now.

Ernest Castorena · Answer

this question is a little less computational. Lee Heavy. Ah, See the information that we want from the problem? They just want us to find the dimension of the north space of a given matrix. Okay, well, let&#x27;s see if our matrix is a negative. 933 negative five. This is a one by two matrix. So it should be out putting five dimensional vector. However, we only have one entry to express leg. The possible solution. That means that five subtract one for therefore, um degrees, which are completely ignored by our matrix. So the null space will definitely be for and the reason why that helped. Because we knew the range waas one. And we know that if we had that I mentioned of the rain and the dimension of the hospice, we should get the total dimension of the cold. I mean,

Runpeng Li · Answer

first we have the Colleen Ami o Space and we first have the basis that is B, which is one minus three t squared, two plus T minus five. He&#x27;s square and one plus two tea. Okay, so the standard days, that&#x27;s just a key one. This one you do is two. Three. Is he squared? So if we transform those constant and call your patient and two squared by by a P want Peter MP three So we have basis be is you want minus three the three. Two p one us. You do minus why p three And he won t o p. Two. So that gives, uh, change. Accorded a matrix from E to see to see to be first. 10 negative. Three and 2165 and 120 It&#x27;s so here&#x27;s dollar matric. So the next thing we need to write t squared as a wiener combination, not the point. Normalcy B. So that is too. Find a p from E to see times record be e and record UFC and work. Even pfc to is the victor 001 and we will solve this system. 121 There were 12 and the three negative. Five and zero and 001 Here&#x27;s the system we need to solve. So we apply the gushing in the nation two. Why don&#x27;t we? I&#x27;ll just write the result here. That is 100 They&#x27;re all one deal 001 and have three negative to one. So that gives, um, Specter record. PB is three negative too. And what? So our we need a combination is that is three times the first in your question one minus three T squared, minus two times a second question two plus T minus five. T squared to us. T minus five teeth were or we had we say t squared. Where you either We can see three and plus one times off one of us to tea, and we can&#x27;t even check. We have, uh, three minus two. Oh, wait. Um, yeah, we have three. Why is two times two is negative for and plus one. So that&#x27;s countenance with zero. And we have that give 60 squared and and positive 10 t squared and has tools when that years. And yeah, we have, uh So then will leave us a T squared and we have to t cancel two by two t like duty here. So that is exactly two D. Oh, Sorry. Uh, second U T squared.

Arsalan Afkhami · Answer

to solve this problem, we can simply compute the inverse off the metrics up then in problem 19. But we can also solve and separately again. So first I write the first victor off basis. See which is one you want one. Then we need to find the victor, which, as the completeness she ran c two, c three that when it&#x27;s multiplied by the victors off basis be gives us the this victor so called to see Juan fly by the first victor 50 plus C two, three year five plus C three, eight to nine on. Then you need to write this stuff equation. And so here is X. Why she wants by the X off the first victor. Plus she too. Bye bye. Except that second on c three, Why would accept that Victor is going to give us the X under the con side. So to see one plus three C to Los age C three is it called one? I was five. She wrong? Plus, we don&#x27;t have the C two for the securities and so I only see three thanks to 36 seas three musical. So negative one, Uh, then five c two minus nine C three musicals. One here you get. We can&#x27;t just simplify. I just want to say Ah see? Want easy call to, um one minus two C three. She lighted by five and then replace. See one here. Then we will have a system of equations. We, uh, two equations. And then again, we do the same. The substitute C two or C three. We have one equation. One, uh, solution. And then we can solve c one c two c three. Other solution would be to use Gaussian elimination. So here we will have an augmented metrics. You have to three eights. There is one and maybe five CEO. You too, Geo. If I maybe nine and here made one and one of what you need to do is to eliminate one side of thematic. So this is the they&#x27;re gonna, for example, we need to most by the first row when number on then added to be said row as to So when it&#x27;s added to the federal get zero on added to the second row. So this is zero. When we do this way, get, uh, she wants C two, C three and the answer, um, for but first equation that you wrote here. He&#x27;s and make to her that we want you letting two bases b your cheese feasts. No face here. This was the first victor. You have two more rectors that we need to so find So a second he&#x27;s to geo one. It&#x27;s a call to see one to negative 50 as before, so it&#x27;s exactly the same. See to 30505 I was three h negative to nine. So here is saying we have It&#x27;s like the same subdivisions. The only difference here. So here we had to see one plus three c two plus 83 c two plus AIDS c three. No, it&#x27;s a call to to and the thing in second month was for maybe five c Ron on this to see three. No, it&#x27;s a call to zero instead on five. See to minus nine c three physical, 21 So assuming on TV solve its by simplifying or using busing and determination to find C one. C two c three maybe gets there you to that&#x27;s to be or soon just on one. So this is the second victor and we have one more so and last form. You want three musical to see one the same as before 20 Last see two, 305 US C three, eight to nine. And here, your assistant of equations that we had above that you have C equations that will give us c one c two c three on They said Victor will be negative. 7 45th 25 So these three vectors that we have obtained are there three columns off the change of basis metrics. So it&#x27;s injured. Metrics from sea to B is equal to fresh. Be right. The first victor now is the second victor that we found. I&#x27;m fair. This is the final answer off this question.

Ernest Castorena · Answer

get for this problem. They want us to find a basis or a subspace of square major cities by two rial entry. The subspace is all matrices with trays. Zero. What do those look like? Well, that&#x27;s right Now and B. C Well, if the trace is going to be zero at this point, we have to have negative. Is this an injury? And now let&#x27;s This is an arbitrary element of our space for any choice of ABC. But now let&#x27;s write our main Chrissy in a different way. Negative. A 00 You can write as the some of these three mantra sees 000 seeing plus 0 +00 be Okay, Well, now we can throughout the scaler. No, since the one ce time 0001 plus spee times +0001 And so we see the this arbitrary element of us has been uniquely written in terms of these three major C and they do form a basis for our subspace. Okay, Awesome.

Problem

In Exercises 11 and $12,$ use an inverse matrix t…

Problem 10 Easy Difficulty

Answer

$\stackrel{\mathrm{r}}{P}_{B}=\left[\begin{array}{ccc}{\mathrm{r}} & {\mathrm{r}} & {\mathrm{r}} \\ {b_{1}} & {b_{2}} & {b_{3}}\end{array}\right]=\left[\begin{array}{ccc}{3} & {2} & {8} \\ {-1} & {0} & {-2} \\ {4} & {-5} & {7}\end{array}\right]$

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$\stackrel{\mathrm{r}}{P}_{B}=\left[\begin{array}{ccc}{\mathrm{r}} & {\mathrm{r}} & {\mathrm{r}} \\ {b_{1}} & {b_{2}} & {b_{3}}\end{array}\right]=\left[\begin{array}{ccc}{3} & {2} & {8} \\ {-1} & {0} & {-2} \\ {4} & {-5} & {7}\end{array}\right]$

Linear Algebra and Its Applications

Catherine R.

Heather Z.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Heather Z.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications