Solution Found!
a. Use coordinate vectors to show that these polynomials
Chapter 4, Problem 32E(choose chapter or problem)
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 \([\mathbf{q}]_{\mathcal{B}}=\left[\begin{array}{r}-1 \\ 1 \\ 2\end{array}\right]\).
Questions & Answers
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 .