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. $$\begin{aligned}4.5 x_{1}-0.5 x_{2} &=1 \\\x_{1}-3.5 x_{2} &=-1\end{aligned}$$

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

determine if the vector v is a linear combination of the remaining vectors $$\mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]$$

Determine which equations are linear equations in the variables \(x, y,\) and \(z .\) If any equation is not linear explain why not. $$2 x-x y-5 z=0$$

(a) In your pocket you have some nickels, dimes, and quarters. There are 20 coins altogether and exactly twice as many dimes as nickels. The total value of the coins is \(\$ 3.00 .\) Find the number of coins of each type. (b) Find all possible combinations of 20 coins (nickels, dimes, and quarters) that will make exactly \(\$ 3.00\)

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. $$\begin{aligned}20 x_{1}+x_{2}-x_{3} &=17 \\\x_{1}-10 x_{2}+x_{3} &=13 \\\\-x_{1}+x_{2}+10 x_{3} &=18\end{aligned}$$

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

determine if the vector v is a linear combination of the remaining vectors $$\mathbf{v}=\left[\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]$$

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. $$\left[\begin{array}{ccccc} 1 & 0 & 3 & -4 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 5 & 0 & 1 \end{array}\right]$$

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. $$\begin{aligned}3 x_{1}+x_{2} &=1 \\\x_{1}+4 x_{2}+x_{3} &=1 \\\x_{2}+3 x_{3} &=1\end{aligned}$$