define-t-mathbbp_3-rightarrow-mathbbr4-by-tmathbfpleftbeginarraycmathbfp-3-mathbfp-1-mathbfp1-mathbf

Question

Define $T : \mathbb{P}_{3} ightarrow \mathbb{R}^{4}$ by $T(\mathbf{p})=\left[\begin{array}{c}{\mathbf{p}(-3)} \ {\mathbf{p}(-1)} \ {\mathbf{p}(1)} \ {\mathbf{p}(3)}\end{array}ight]$
a. Show that $T$ is a linear transformation.
b. Find the matrix for $T$ relative to the basis $\left\{1, t, t^{2}, t^{3}ight\}$ for $\mathbb{P}_{3}$ and the standard basis for $\mathbb{R}^{4} .$

Runpeng Li · Accepted 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 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

Millie Lopez · Answer

Hi. So, for this exercise, we need to find the Matrix for the transformation off T By the given basic vectors that artie t one t two we buy knowing that we can start by finding the images off each one of them after their leaner mapping anti. So we start by Do you want equals to t He&#x27;s gonna have a multiply by there basic vectors that it&#x27;s one t anti too because of we&#x27;re looking sorry. So we&#x27;re looking for t one. Our values are going to be centered on one. So one is going to be one t is going to be Cyril. NT two is going to be also Ciro Ang. These values, we can change them too. A Ciro A one and a two. Okay, so in our general equation, we&#x27;re gonna be So did I in each one of them for defining off the leaner mopping. So we start by tree a Cyril, we have the brother that he&#x27;s one. So we right here, One applause we have here 50 So five also gonna be multiply by one. But then we have minus two one minus two a one so minus two, we have the value off a one that is zero. So we ride that to its multiply vice zero later. We right here that they have tea. These right here. Then we put a clause for a one that we have for a one. A one has the value of zero. So for want to buy Vice zero waas 82 That also has a value of zero. So we right here zero close it and we right t two Now we go tree stays the same. So east tree Here we have five minus two buddies zero So here we ride five, but its multiple righty So is five t here we have C room and Syria also here. So this is going to be more to buy vice zero. So this is going to be zero to you in leaner mopping. This is right then, like tree five. This zero. That&#x27;s our first call him. So you go for our second that we&#x27;re now gonna use that t won t t Sorry. You write the equation that equals two t one t t two. So we now want to find a value off t. So now our one is going to be 80 Our T is going to have the value of Warren anti to also going to be zero. So now we do. The same thing is going to be our 80 a one and a two We started. We write the equation, so he&#x27;s going to be t t equals toe tree 80 waas five a Cyril minus to a one old is wanted by by t one applause four a one pose a to And this is more to buy by tedium. Okay, so we now simplify it by we have the value that treat one 380 is going to be zero. So we right here tree zero five a zero Also gonna have a multiplied by C room to a one these time is going to have the value of one. So we multiply by one and we write our tea. A one for a one also has a value of one. So he&#x27;s going to be four. Don&#x27;t supply by one applause A to also has a value of zero. So we write zero and we hear Multiply by t two. Now we go here has the value of zero. Here we have the value of zero. But here we have negative too. So is going to be negative to here we have four. And here we have zero. So we&#x27;re going to write it like zero minus to T plus for T two in a linear mapping. This is also right and lie zero negative too. And four, then for our next one is going to be t equals t two. So this is going to be able to t multiply by one t t two. This time we want to find t two. So one gonna have a zero and t is gonna have a zero, and teacher is gonna have a one. So we right here zero zero one. These are gonna be our values for they want a Cyril a one on eight to So let&#x27;s start with the equation again. Three tree a zero waas five a zero minus two a one all multiply by t applause for a one because a to and this is going to be multiply by t two. Okay, so now we served the type the values a Cyril&#x27;s gonna be here. Ciro a serum again Here. Zero a 10 a one savvy Cyril here is gonna be a to one. So we right here, three won&#x27;t by by zero five multiply by zero negative to want to buy also via zero, for it&#x27;s gonna be also multiple. I zero and we have here alone. Plus want It&#x27;s now the value that this is gonna give us this is gonna be a zero is gonna be here, Cyril. And here&#x27;s zero. So we right here, Ciro plus zero t we have here that is going to give a Cyril but is plus one. So one multiply by t two. It&#x27;s gonna be plus one berated like this so it could be different. T. Teoh, this is writing in Poland and Lander. Let mapping is gonna be zero Cyril and one Now we have all the values we have. This one. We have this one and we have that one. So now we&#x27;re gonna with them build our matrix. So now we write the basic vectors coordinates as the columns for our matrix told that May right here Matrix is gonna be am And then it will be then basic matrix for the linear transformation is going to be equal to do you want T t and t t two. So on t one we had the values off tree five and zero on t t. We had the value off zero negative too. And four on t two. We had the values off zero zero and one and is going to pay the final answer. Oh, my God. This is going to pay the final answer. Thank you.

R M · Answer

a broken 14? Yeah. Coach in pain reception or be Is it with you? Want one you want to? He wants cleaning e two n word Any to To into three. And if anyone seems to be to and in me screen and seeing is this article one. You&#x27;ll notice that you you went one is April 18 year old eating two people to you are if being between you it But, uh uh What reads? Well, it one word G people through the trees for a two to just equal through the trees. Oh, easily, which is equipped toe. Ah, sure. Nutty. Our Jane is equal toe the trace. Our e r. Jane is April Prusiner. We&#x27;re on and Jean B wrongs to know one. Oh, it&#x27;s not equipped to to show that the G ands on any combination of elements off. See? So, um, you you want one? It&#x27;s equal to one time and one. And the, uh, you weren&#x27;t is equal to zero times Warren and t he want is equal to you know nothing. You all into one. It would Did you find this one? You are eat your total. They will hunt findings. One being beginning with B. They put two uniforms, one senior role. Eve. When you warn the sequence, most times one to me girl, you sitting toe is April 12. You know, time is one people you need to be being gave birth to one finds one. So you see these coefficients We can write what magic saw. 15 being being believable one, you know, And you and your but one. You know you, you know, And one and the next tricks overeating for me, it&#x27;s equal to to really negative seats. See? You know, one were negative for zero, you know, anarchy to gets. Ah, the magics. Uh, that&#x27;s inspirational. A for seeing we have to mark the blind means much diseases with each other and the fund of those runs. We mean it was toe. So we&#x27;re 50 or you seeing is different. Three, uh, McCutchen be as d o a. It&#x27;s even though that trains off a she&#x27;s people through the trees are the ones you negative seats or And you you Negative. She&#x27;s here with us for when? So here, But saying bad here

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.

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

Problem

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

Problem 10 Medium Difficulty

Answer

a) $T$ is a linear transformation.
b) $=\left[\begin{array}{cccc}{1} & {-3} & {9} & {-27} \\ {1} & {-1} & {1} & {-1} \\ {1} & {1} & {1} & {1} \\ {1} & {3} & {9} & {27}\end{array}\right]$

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a) $T$ is a linear transformation.b) $=\left[\begin{array}{cccc}{1} & {-3} & {9} & {-27} \\ {1} & {-1} & {1} & {-1} \\ {1} & {1} & {1} & {1} \\ {1} & {3} & {9} & {27}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Kayleah T.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Anna Marie V.

Heather Z.

Kayleah T.

Samuel H.

a) $T$ is a linear transformation.
b) $=\left[\begin{array}{cccc}{1} & {-3} & {9} & {-27} \\ {1} & {-1} & {1} & {-1} \\ {1} & {1} & {1} & {1} \\ {1} & {3} & {9} & {27}\end{array}\right]$

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

Linear Algebra and Its Applications