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Assume the mapping $T : \mathbb{P}_{2} \rightarrow \mathbb{P}_{2}$ defined by$$T\left(a_{0}+a_{1} t+a_{2} t^{2}\right)=3 a_{0}+\left(5 a_{0}-2 a_{1}\right) t+\left(4 a_{1}+a_{2}\right) t^{2}$$is linear. Find the matrix representation of $T$ relative to the basis $\mathcal{B}=\left\{1, t, t^{2}\right\}$

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

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 4

Eigenvectors and Linear Transformations

Vectors

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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's gonna have a multiply by there basic vectors that it's one t anti too because of we're looking sorry. So we'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'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'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's our first call him. So you go for our second that we'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'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's going to be four. Don'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'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'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'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't by by zero five multiply by zero negative to want to buy also via zero, for it's gonna be also multiple. I zero and we have here alone. Plus want It's now the value that this is gonna give us this is gonna be a zero is gonna be here, Cyril. And here'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'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'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.

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