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The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ Find the coordinate vector of $\mathbf{p}(t)=1+4 t+7 t^{2}$ relative to $\mathcal{B} .$

$p_{\mathcal{B}}=\left[\begin{array}{c}{2} \\ {6} \\ {-1}\end{array}\right]$

Calculus 3

Chapter 4

Vector Spaces

Section 4

Coordinate Systems

Vectors

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In mathematics, a vector (…

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Let $\mathcal{B}=\left\{\m…

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okay. In discussion. Settle cases B is equal is given one plus t square, deep, lusty square and one plus two t plus thes. We're okay and is the basis off p two and find the coordinate vector off. Beauty equals to one plus 40 plus 70 square related basis. Okay, now can. Now we can write this in metrics. Thomas de Squire tea and one so we can write this. Let one from this day, Squire. First, we have to write this T square one in the coefficient off zero and constant one. Now, from this P square is one. He is one and question zero from this D Square one, the coefficient off piece, too. And constant one. And the solution is seven, four and one. Okay, step by. Step on. We have us all this now, even swallow this. Okay? So fast. We have to make this part zero. So we will do. Are three r three equals toe one minus are Okay, So what we get is 111 sailing 0124 on 0106 Okay. And now now what we do on one needle? No, again. They will do this portion zero. Okay. And for this, we will do R two equals toe are two minus artery. So we get 111 70 0 to 4 minus secrets minus two on this will be same. 0106 Okay, now what we do, you do simply now we will find out, uh, are to divide by go only Why get 111 seven 00 You were too. Is one minus one on a 010 six. Okay. No, What we do is r one equals toe are one minus R minus artery. For that, they will get When Zito Zito 2001 minus 10106 Okay, so now we 13. Our goal with our three. So what? We get 1002 0106 on 001 minus one. Now we have the standard metrics. Okay? And what is our answer? This is over. Answer. The answer is toe six minus one. P. B equals two. This is our final answer. Thank you.

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