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Question

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}ight\}$ be a basis for a vector space $V$ and $T : V ightarrow \mathbb{R}^{2}$ be a linear transformation with the property that
$$T\left(x_{1} \mathbf{b}_{1}+x_{2} \mathbf{b}_{2}+x_{3} \mathbf{b}_{3}ight)=\left[\begin{array}{c}{2 x_{1}-4 x_{2}+5 x_{3}} \ {-x_{2}+3 x_{3}}\end{array}ight]$$
Find the matrix for $T$ relative to $\mathcal{B}$ and the standard basis for $\mathbb{R}^{2} .$

Runpeng Li · Accepted Answer

Okay, so for problem for I r. Again given assumption we have t of x one, be one of those x two you two plus x three The three It&#x27;s the new vector, which is to x one minus four x two was five like three and ActiveX too. Plus three x three. So to find out the matrix for tea rented to be first need to find out the f B one from court in a C where C is Thea faces off off are too. So here we can just take X one to be one and next to x three to be zero. So our specter will be too zero. And similarly, we can find bee the F B two. So this will be negative for Nick to one and finally t f B three so T o b three Just be five and three now. So the Matrix TB, we&#x27;ll just be the span off these three vectors. Two negative four and five zero negative one and three

Runpeng Li · Answer

pick. So, in this problem, we&#x27;re already given the, um, Helene Alina transform Matrix relative to be, which is a three by three matrix. That is the real inductive. 61 05 elective one one negative two and seven. Now the only thing we need to do to find out a t of three of the one minus for B two is to take the coefficient off the given victor three V one minus four Be too. So it&#x27;s three off the one negative for P 20 for B three and enter the cleaner transformation off this matrix and we find that is 24. Order for century and active 20 for a second entry and three waas eight. You love it for the second entry. So that is actually the tea. Uh, three three b one. Honest for Pete to

Runpeng Li · Answer

okay for problem Problem 10. We have the leaner, faster mission maps from P three to Arthur are poor. Such that t of pulling nominal is equal to by taking different input to the Polo Meo. Just pee of connective three first p A connected one and p of one and p up three. Hey, So the first thing we need to show that is that, um to show this transformation is linear. Okay, so do that. My transformation. We can first check tee up. People ask you where panic you are all clean on yourself. Order to be so then, by our assumption, we can have a p rescue while we take the include to be elected three. And he ask you with the input elective born people ask you with include one and people ask you with your food three. Now, because we are adding these two point nominees so we can we can calculate the number Sweet that with this input on dad them together just just by by the rules, our tradition. So that&#x27;s he, uh, 93. I trust you collected three and PR. Negative one. Uh, you, uh, negative one and p of one less queue of one. And here, three askew on three. So by now we can separate this, uh, this vector with by a vector off Minami O P and the vector polynomial Q. So it turns out to be thio he that&#x27;s t l Q. So that&#x27;s the first part here. So the second part is considered skater Linda times people. So again, by our assumption, we still have Lunda cons. P off negative three Lambda SPF connective one, huh? Times p of one and Linda Times p of three. Excuse me. So we can take out the skater from our rector&#x27;s. So we have loved a in front of our specter and P uh, negative three. He, uh, connected one he won and ps three. So that turns out to be Lunda time. So? So we&#x27;re done for the first part because I our this relation and second relation. Then we can conclude that he&#x27;s a renewed transformation. Now, Part B. We need to find the matrix for tea relative to the basis won t t square t t cubed for p three and standard basis are for Excuse me. So now let&#x27;s, um defined business P start this is B to be given basis in our assumption, which is one two he squared and t Q. So we first calculate, uh, actually the tea with the input up won t t square and teeth cute separately. So first we have to be one which gives our vector all once and then we&#x27;re plugging t which gives Dr Inactive three negative one, one and three. Then we&#x27;re plugging our t squared. This gives Excuse me. Um 911 night, and at last we plugging our thank you. So we had a vector collective 27 negative. 11 and 27. So our metrics will be this band of these four vectors. So this will be he of this is B should be one collective three nine Next 27 one defective. 111 1111 and 139 27. And that&#x27;s it.

Runpeng Li · Answer

Okay, so for problem to we have the basis di at his d one and you too, and are so basis be, which is B one B two. So we we need to find out the matrix t relative to D and B. So by the giving permission ti of the one is to be one minus three b two and t of d to its inactive for B one plus five b two. So by using that relation by observing their coefficient, you can have a transformation off the one and we should have a deep coordinate. Sorry, the be coordinate here. So the first thing that the 100 people made by our first equation it&#x27;s okay to be one Sorry. Step two. We should be two and three and would be other. You too. And under ordinate b, this will be by our second equation, like a born by now the meet our metrics t I would just be the span of these two vectors to elective three next four and five

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.

R M · Answer

a home number 12. Ah, by fear and and my theory, um, 6.45 that the a see is equal toe. Okay. Upsy and being a, we&#x27;ll be so see, uh, anyone one is able to Sometimes it&#x27;s one when it&#x27;s your clothes, you know? And, you know, 13 times a year. I knew that the one liners here? Yeah, pretty, which is equal to 20 and that they weren&#x27;t ending it. You&#x27;re a D or e wanted to. It is equal to two times. You know why this one will see you. The little military there. What? Three times the euro Negative zero minus one. Lost. You know, Thomas, three times zero. She&#x27;s equal to negative one. See you. You know, and get support. We&#x27;re dio into one. This is a quinto times, you know, minus zero bus, Do you know? And, you know, negative. 03 times zero Needed to zero minus zero love three times more, which is able to 000 and three or the, uh, e doing too. Since April 2. Time sealing linens. You. That&#x27;s one. You know, you need to, you know, many times more minus C Rose wine. It&#x27;s near. What a times they&#x27;re sorts. It wont 13 he runs. So now we need to show that bees t OBY 111 went toe and to wonder, too, as they learn the senior combination off that from C to ah t even one can be written as and three times E 11 minus. One times he went to plus New York Times E to one minus, but it&#x27;s one times into you are They want to. It&#x27;s equal to negative one times 11 plus zero times even to plans, you know, fine each one when it&#x27;s one times e to Teoh e toward. It&#x27;s equal to, you know, times even when one plus you times he went to. Thus zero times detour breath three times e t uh, into two. It&#x27;s equal to one times you want one 13 year times even to plus you know time 8 to 1 plus two year old times a to So now we right Uh, the e coefficients interned a vector off see so accordingly and using the cuttlefish is we can write with geometric. It&#x27;s which is equal to to negative one. Do you know that negative. Warm negative. 100 and negative. One 0003 13 You know NZ? No. Is that amen? Tricked. So be it&#x27;s April cruise director. Negative seven toe war ending three. So Ah t Oh, a We&#x27;ll see. But you went to T maj. It&#x27;s or CMB times a metrics for me. So it&#x27;s so now we&#x27;re trying toe motto, floppies, many trips. I can start the game. So to negative one. Do you know anything at the one you get to warn? See room here on anything. One zero. You you know, 13 You loons. You. No, really. Why? Negative seven to one and negative three. So it&#x27;s equal to negative 19. Negative too, you know, and and big to, ah, even a is able to no need to 19 0 negative toe. And eight no good should be, as we know that you okay, is equal. That transformation. Uh, negative. One tool and negative. 30. So is he Went to your plans? A minus two negative. 30 zero seven. Yeah. Three times negative. 30 seven. Minus tomb plus three times. One which will be finally going to the same here. Negative. 19 0 negative to and eight

Problem

Let $T : \mathbb{P}_{2} \rightarrow \mathbb{P}_{3…

Problem 4 Easy Difficulty

Answer

$\left[\begin{array}{ccc}{2} & {-4} & {5} \\ {0} & {-1} & {3}\end{array}\right]$

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

Linear Algebra and Its Applications

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$\left[\begin{array}{ccc}{2} & {-4} & {5} \\ {0} & {-1} & {3}\end{array}\right]$

Linear Algebra and Its Applications

Catherine R.

Kayleah T.

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.

Kayleah T.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications