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Let $\mathbf{p}_{1}(t)=1+t^{2}, \mathbf{p}_{2}(t)=t-3 t^{2}, \mathbf{p}_{3}(t)=1+t-3 t^{2}$a. Use coordinate vectors to show that these polynomials form a basis for $\mathbb{P}_{2} .$b. Consider the basis $\mathcal{B}=\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$ for $\mathbb{P}_{2} .$ Find $\mathbf{q}$ in $\mathbb{P}_{2}$ given that $[\mathbf{q}]_{\mathcal{B}}=\left[\begin{array}{r}{-1} \\ {1} \\ {2}\end{array}\right]$

$q=\left[\begin{array}{c}{1} \\ {3} \\ {-10}\end{array}\right]$

Calculus 3

Chapter 4

Vector Spaces

Section 4

Coordinate Systems

Vectors

Johns Hopkins University

Oregon State University

Harvey Mudd College

University of Nottingham

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In mathematics, a vector (…

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Let $\mathcal{B}=\left\{\m…

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The set $\mathcal{B}=\left…

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Let $\mathcal{B}$ be the b…

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Let $\mathbf{v}_{1}$ and $…

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Assume the mapping $T : \m…

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Let $p_{1}(x)=1+x, p_{2}(x…

Yeah, in this trouble were given a set up. So we'll start solution by writing the organ directors for each polynomial. So for people in the quarter, Inspector Ho. But like this 101 said, the 1st 1 is the question. But these year old 2nd 1 is to one of you two. So using this Pete, too apartment that too perfect. You could be rich. Zero. What being? And for P three, that could be, um, that would be one one negative three. So then the augmented matrix phoned my bees. Corn factors would be 101 01 negative three and 11 Negative three. Now let's write this over mental midgets in traditional echo form. And that is he could do warn. 00010001 is you can see. We have three columns and each column has a few bits elements. So three columns, three events elements at each column. So it means that these vectors Spector's spend our fee. It means that the given Colonials no meals spin he two. Now in part, we were given 1/4 collector for a loan Jew, and that is next to 112 And whereas to find this bologna to using this So we know that cube of native one times people plus one p two plus thio three So Dunn will be negative one times ones You're one plus one times zero wounding three waas Two times One wonder three And we find this to be one the next 10. And using this, uh, using the information that these numbers represent corruption tease. We can ride the roller milk you is one plus B t minus 10 t spring.

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