91Ó°ÊÓ

Apply Jacobi's method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you like. $$\begin{aligned} 7 x_{1}-x_{2} &=6 \\ x_{1}-5 x_{2} &=-4 \end{aligned}$$

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 0 & 3 \\ 0 & 1 & 0\end{array}\right]\)

Determine which equations are linear equations in the variables \(x, y,\) and \(z\). If any equation is not linear, explain why not. $$x-\pi y+\sqrt[3]{5} z=0$$

Determine if the vector v is a linear combination of the remaining vectors. $$\mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r} 2 \\ -1 \end{array}\right]$$

Determine which equations are linear equations in the variables \(x, y,\) and \(z\). If any equation is not linear, explain why not. $$x^{2}+y^{2}+z^{2}=1$$

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. \(\left[\begin{array}{rrrr}7 & 0 & 1 & 0 \\ 0 & 1 & -1 & 4 \\ 0 & 0 & 0 & 0\end{array}\right]\)

Determine if the vector v is a linear combination of the remaining vectors. $$\mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r} -1 \\ 3 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r} 2 \\ -6 \end{array}\right]$$

Apply Jacobi's method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any direct method you like. $$\begin{aligned} 2 x_{1}+x_{2} &=5 \\ x_{1}-x_{2} &=1 \end{aligned}$$

Determine which equations are linear equations in the variables \(x, y,\) and \(z\). If any equation is not linear, explain why not. $$x^{-1}+7 y+z=\sin \left(\frac{\pi}{9}\right)$$

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. \(\left[\begin{array}{llll}0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1\end{array}\right]\)