a. Use coordinate vectors to show that these polynomials

Chapter 4, Problem 32E

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QUESTION:

In Exercises 27–30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

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 q in \(\mathbb{P}_2\), given that \([\mathbf{q}]_{\mathcal{B}}=\left[\begin{array}{r}-1 \\ 1 \\ 2\end{array}\right]\).

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QUESTION:

In Exercises 27–30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work.

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 q in \(\mathbb{P}_2\), given that \([\mathbf{q}]_{\mathcal{B}}=\left[\begin{array}{r}-1 \\ 1 \\ 2\end{array}\right]\).

ANSWER:

Solution 32EStep 1 Consider the following polynomial: The coordinate mapping produces the coordinate vectors with respect to standard basis, are respectively.a)The objective of the problem is to show that the given polynomial form a basis for .To show the set of polynomial form a basis for , construct a matrix with the coordinate vectors of polynomial as the columns of the matrix.The required matrix is .

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