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In $\mathbb{P}_{2},$ find the change-of-coordinates matrix from the basis $\mathcal{B}=\left\{1-3 t^{2}, 2+t-5 t^{2}, 1+2 t\right\}$ to the standard basis. Then write $t^{2}$ as a linear combination of the polynomials in $\mathcal{B}$ .

$t^{2}=3\left(1-3 t^{2}\right)-2\left(2+t-5 t^{2}\right)+(1+2 t)$

Calculus 3

Chapter 4

Vector Spaces

Section 7

Change of Basis

Vectors

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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's square and one plus two tea. Okay, so the standard days, that'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's so here'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's the system we need to solve. So we apply the gushing in the nation two. Why don't we? I'll just write the result here. That is 100 They'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'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'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.

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