in-exercises-11-and-12-find-the-mathcalb-matrix-for-the-transformation-mathbfx-mapsto-a-mathbfx-when

Question

In Exercises 11 and $12,$ find the $\mathcal{B}$ -matrix for the transformation $\mathbf{x} \mapsto A \mathbf{x},$ when $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}ight\}$
$$
A=\left[\begin{array}{rr}{3} & {4} \ {-1} & {-1}\end{array}ight], \mathbf{b}_{1}=\left[\begin{array}{r}{2} \ {-1}\end{array}ight], \mathbf{b}_{2}=\left[\begin{array}{l}{1} \ {2}\end{array}ight]
$$

Chris Trentman · Accepted Answer

were given a Matrix A and column vectors B one B two, which we&#x27;re told me when in B two form the basis Be and a Is the Matrix for a transformation from X to a X and were asked to find the bee matrix for this transformation. So to do this, we&#x27;re going to take p to be the column vectors. Be one matrix of column vectors. B one b two. It&#x27;s The Matrix with Colin Victor&#x27;s to negative one in 12 Then we have following the example in the book Be Matrix is given by P in verse a p. We can calculate p inverse pretty easily using p, using our formula for the inverse of the two by two matrix. So the determinant of P is four minus negative one, which is five. So we have 1/5 times I&#x27;m sorry times and then we have Well, we&#x27;re going to want to switch D and A so we have to and then negative be so negative. One negative sees a positive one, and then a so too times. And then we have our matrix A which were given us matrix of column vectors. Three negative 14 negative one and finally, p again. So to negative. 112 And so I get 1/5 times to negative 112 and then calculating a P. I get three times two is six minus four is too. Three times one is three plus eight is 11 and negative too. Plus one is negative. One negative one minus two His negative. Three 1/5 times multiplying again. I get four plus one is five. 22 plus three is 25 tu minus two a zero and 11 minus six is five. And so distributing the scaler I get the matrix one five 01 And so this is Thebe matrix for this Trans.

Runpeng Li · Answer

Okay, So in this problem, we only need to apply our the&#x27;re, um your mate, you know, textbook. So it says if we have the Matrix P Well, in this case, our matrix peas. Just a span, not B one B two. So Matrix B. We know this matrix. Then our bi matrix will be peeing burst. Time say times P. So we just need to find out the product. So, Pete, members, by my calculation, I won&#x27;t do this step by step. Just a house. Just write down this of embers. It&#x27;s 11 I think the 23 now Matrix A is next to one for next 2 to 3 and matrix P. It&#x27;s just a three negative 1 to 1. Okay, So as first find, it was too part of the first product. So first entry, Nick one times. Like to warm. Plus two two times one. So that&#x27;s negative. Three and one plus one times four plus one times three. So that&#x27;s seven and two minus six. So it will be negative for And the last entry we have connective aid. That&#x27;s nice. So it&#x27;s one now, three negative. 1 to 1. Okay. Now for this to matrix. We&#x27;ll search for these two missions, since we have next of three times three active and I last 14. So it will be five and three. Ah, seven. So he&#x27;s 10 and next to 12. Us, too. So selective. 10. And we have four plus one. So he&#x27;s five. No, just put the skater into our matrix. So we have one too connected to and one So this is

Runpeng Li · Answer

Okay, so for our problem 30. Um, we&#x27;re given three by three matrix, which is sector 14 for active 14 negative. 33 nine negative. 31 11 4 11. And for the basis, which is B one B two B three. Now we have. Now we know that our matrix p she&#x27;s the span of these three vectors. Negative one Negative too. One and active. One elective. Want one you wanted to. And a zero. Okay, So to find the matrix speak bee matrix. So that&#x27;s just a d. We need to on p murders times eight times, people. Now I is this, uh this is three bedroom, three drinks. You want imputed alien computer ubers? I had these kind of hard, so I I did this in Wolfram, So Oh, just remember that. I mean, the whole empire dissolution of this just to make the numbers off a B times eight times e and or it will be hate. Three sticks, 013 And so it was a wrong number three and

Bobby Barnes · Answer

So if we want to solve this by an augmented matrix, let&#x27;s go and get the augmented matrix set up first. So remember, we&#x27;re trying to get this into the form of X as it would be. And remember, we just kind of pretend like the X isn&#x27;t there and the equal sign. So first we will rewrite a so be 1 to 1 negative three negative. 12053 And then for this equal sign, we just put this bar going straight up and down. Then we write, be on the other side would be one or 01 negative one. Now, if you&#x27;re allowed to use a calculator to do this road reduction, Um so normally it&#x27;s like the r A f and a calculator. This should give us the matrix 111 and then a bunch of zeros here and then on the right side, it is 0.6 negative. 0.8 and one. I remember the way we read this is X one is equal to 0.6 x two is equal to negative 0.8 x three is ego one. And since we want this as a vector, we would just say X is equal to 0.6 negative. 0.8 one. So pretty straightforward if you can use a calculator. But if for whatever reason you can&#x27;t we can still do this by hand. Uh, pretty easily. So let&#x27;s go ahead and pretend like we didn&#x27;t do that and just kind of scoop this down, but at least we know like it. For some reason, we don&#x27;t get this. We may have done something weird with one the calculations and will give us a reason to, like, go back and check. But what I would do to start is so this is a non zero number so we can keep it there and then remember, we want to clear out everything below. So third row already has zero there, so we just need to clear off the three. So if we multiply row one by three and then add that to row to then that negative three councils out. So first two rows stay the same. So 1 to 1 zero third row 053 negative one. Now this negative three becomes a zero. So two times three plus negative one is 51 plus three plus two or one times three plus two is 50 times three. So that doesn&#x27;t change the one there right now. We could divide all this by five, but notice that if we actually just did row one are negative road are not negative room one negative road to plus row three, then the five below it cancels out. So let&#x27;s go ahead and do that. So remember, first to roast stay the same because we haven&#x27;t done anything with us. A 1 to +10 0551 And now this is still going to be zero. That is zero because it cancels out and then negative five plus two is negative two and then negative one plus negative one is negative. Two All right. And now we can go ahead and divide this first row here by negative one half. So we do. Negative one half wrote to herbs are route three. So first few rows once again stay the same And now the second row. But we just divide everything by negative too. So that would be 0011 So we again get x 301 So it looks like We&#x27;ve done everything good so far. Now we want to clear the ones above this, or I should say the numbers above this. So there is actually a wonder. And to do that, um, to clear row one, we would just need to subtract. So it be Row three plus row or negative Growth three plus road one. So let&#x27;s go ahead and write that out first. So the one and the to stay the same, that becomes zero. And then negative one plus zero is negative one now for row to we would need to do negative five row three plus road one. And so that stays the same. The five stay the same. The second five cancels. And, uh, sorry. Now we were doing negative five or three times. One eso I&#x27;d be negative five, and then we add that toe once. That gives us negative four and then the last row stays the same. So it&#x27;s 0011 Right now. Let&#x27;s go ahead and clear of this five in row two. So we&#x27;ll do 1/5 Row two. And that means the first two rows stay the same. So 1 to 0 negative one 0011 and that would be 010 and then negative for over five. That&#x27;s negative. 0 20. So the second one looks good as well. And then we just need to clear this, um, to above are one. And doing that, we would get eso We need to do negative, too. Row two plus Row one. And so the 2nd and 3rd rose stay the same. So we have 010 negative. 0.8 0011 And then if we clear going up so I&#x27;d be 100 and then negative 0.8 times negative two is 1.6. Add that to negative one, and that gives us 0.6. And so once again, this says X one is equal to 0.6. X two is equal to negative 0.8 and x three is the Goto one which says our Vector X is going to be 0.6 negative 0.8 and one. So this is our solution. And let&#x27;s just double check to make sure that matches up with what we got the first times. Yeah, 0.6 negative 0.8 and one say again if you can use the just matrix to do it, Um, that&#x27;s normally the way to go, because doing these calculations by hand is very easy to make a mistake. But if you also have to, like, show all your work, then we also cover our bases doing that. But we used it to double check so

Aymara Gallardo · Answer

Okay, so we have to prove that the Matrix A is a merciful being. So we have to protect a times B is identity and the other way around. So let&#x27;s make that brother first. That&#x27;s a Now be okay. So we kept negative Two friends. One that&#x27;s negative. One class three have stems to this three. So they keep us three minus two Now on the bottom, which have negative six, three cups, things for the makes six and end up here. We have one minus one on the bottom. We shall city minus two. And that is exactly then ended metrics. Okay, this calculate now be James a See if you can fit in here. Any of you to 13 Perhaps nudity won&#x27;t have. Where to three. What? Okay, what do we have? It&#x27;s gonna be the first coordinate. Negative to last three in the second quarter, earnings get gonna have. Did he have minus three halves? We have here negative for less poor and on the bottom. We have six over too. Rhinos for over two you see in the first quarter than it would have one in the bottom. We have sirrah, Sirrah over here. And now this is too over to. That&#x27;s actually one. And so we&#x27;ve verified then, ese. Remember Soapy

Runpeng Li · Answer

problem. Eight. So we first right down the vectors. The one is connective. 18 He too is one negative five and be three iss. Sorry. I see one C one ISS 14 and see to is one. So we let&#x27;s first find the, uh, change of coded matrix from me to see. So in order to find the change according the Matrix from B to C, we put column vector See, on the right hand side 1411 and I call Alexis would be electable. 18 one, activate and goodbye. So we need to turn the left inside to be the ad in the Matrix. So excuse me. We performed the guttering ammunition. First thing we do is too used a second role, minus four times out. First role. So 11 if you want one. Four times. Four times four. You know, one time, one minus four would be negative. Three, eight miners. Negative for So that is 12 and negative. Five minus four. That&#x27;s a negative. Naughty. Okay, so next we used the first roll. Plus three. Happy about the second roll. We can do the three having a second row purse because we to turn, um, this entry to be one. So Israel, we&#x27;ll keep the same 11 and negative one and one negative one. And the one and zero Nothing. One, four and negative three. So we add first row by the second rule. So that would be one, you know, And three and negative too. And zero, we divided. Divide both or multiply both sides by tactical one or two by a second row bite into one. That&#x27;s one before. And so our change of corny matrix from you to see is the right hand part 32 naked before and three. All right, so to find the numbers, C two b. That&#x27;s, uh that&#x27;s to say, um, we need to find it Inverse off this matrix greed and to do next for Nick on three. So do this naked, too, Before and three and 100 one. So first thing I want to do is too. Divide the first roll by three. This one Now the negative to over three and 1/3 and zero. And we are So, um, have second role next, four, three, zero and one. So that is should be, uh, a narrow here. So we divide the second row by four. So we have next one new quarter and zero on 1/4. So one night get to third and 1/3 and zero. Okay, so So we add the first roll by the second roll awake, we add the second anniversary. So that&#x27;s zero and 3/4 3/4 minus to third. This will be nine divided by 12 minus. So that&#x27;s eight, if I did by 12. So that&#x27;s 1 12 So 12 right on DDE, we still have 1/3 and 1/4 and first role is one negative to over three. 1/3 and zero. So as first turn turned the second entry, that&#x27;s money. Sorry. It&#x27;s just multiply the second roll by 12. So we have 01 and for and three and one to over three and 1/3 and zero. So we we use the first roll had negative, too at 2 32 3rd off the second row. So we have one zero 01 and 1/3. So that&#x27;s so 1/3 plus to third up. Four. So that&#x27;s eight there. Yeah, so that&#x27;s 1/3 and we have three. Oh, it should be on the first rule. We have three and then we have say to third, all three will be too. And that by a zero said, That&#x27;s too. We have 0143 So this is our change of holding the Matrix and from Did you see? And that is 32 or three.

Problem

In Exercises 11 and $12,$ find the $\mathcal{B}$ …

Problem 11 Easy Difficulty

Answer

$\left[\begin{array}{ll}{1} & {5} \\ {0} & {1}\end{array}\right]$

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$\left[\begin{array}{ll}{1} & {5} \\ {0} & {1}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Samuel H.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Catherine R.

Heather Z.

Samuel H.

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

Linear Algebra and Its Applications