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In Exercises 5–8, find the coordinate vector of x relative
Chapter 4, Problem 7E(choose chapter or problem)
In Exercises 5–8, find the coordinate vector \([\mathbf{x}]_{\mathcal{B}}\) of x relative to the given basis \(\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}\).
\(\mathbf{b}_{1}=\left[\begin{array}{r}1 \\ -1 \\ -3\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}-3 \\ 4 \\ 9\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}2 \\ -2 \\ 4\end{array}\right], \mathbf{x}=\left[\begin{array}{r}8 \\ -9 \\ 6\end{array}\right]\)
Questions & Answers
QUESTION:
In Exercises 5–8, find the coordinate vector \([\mathbf{x}]_{\mathcal{B}}\) of x relative to the given basis \(\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}\).
\(\mathbf{b}_{1}=\left[\begin{array}{r}1 \\ -1 \\ -3\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}-3 \\ 4 \\ 9\end{array}\right], \mathbf{b}_{3}=\left[\begin{array}{r}2 \\ -2 \\ 4\end{array}\right], \mathbf{x}=\left[\begin{array}{r}8 \\ -9 \\ 6\end{array}\right]\)
ANSWER:Solution 7EStep 1 Consider the following vectors and Our strategy is to find the coordinate vector of relative to basis Recall the fact that “to find the coordinate vector of relative to basis , solve the equation for , and then write them in column-wise”In this case, and As , get …… (1)This equation can be solved by row operations on an augmented matrix.