91Ó°ÊÓ

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

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

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

Apply Jacobis 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. $$\begin{aligned}3 x_{1}-x_{2} & =1 \\\\-x_{1}+3 x_{2}-x_{3} & =0 \\\\-x_{2}+3 x_{3}-x_{4} &=1 \\\\-x_{3}+3 x_{4} &=1\end{aligned}$$

Determine if the vector v is a linear combination of the remaining vectors. $$\begin{array}{l} \mathbf{v}=\left[\begin{array}{r} 3.2 \\ 2.0 \\ -2.6 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1.0 \\ 0.4 \\ 4.8 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r} 3.4 \\ 1.4 \\ -6.4 \end{array}\right] \\ \mathbf{u}_{3}=\left[\begin{array}{r} -1.2 \\ 0.2 \\ -1.0 \end{array}\right] \end{array}$$

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

Determine which equations are linear equations in the variables \(x, y,\) and \(z .\) If any equation is not linear, explain why not. $$(\cos 3) x-4 y+z=\sqrt{3}$$

Find a linear equation that has the same solution set as the given equation (possibly with some restrictions on the variables.) $$2 x+y=7-3 y$$

Determine whether the given matrix is in row echelon form. If it is, state whether it is also in reduced row echelon form. \(\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\)

Determine if the vector b is in the span of the columns of the matrix \(A\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 5 \\ 6 \end{array}\right]$$