The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. displaystyle{left[begin{matrix}{1}&{0}&-{1}&-{2}{0}&{1}&{2}&{3}end{matrix}right]}

Let $AX=B$ be a system of linear equations, where A is an $m\times nm\times n$ matrix, X is an n-vector, and $BB$ is an m-vector. Assume that there is one solution $X=X0$. Show that every solution is of the form $X0+Y$, where Y is a solution of the homogeneous system $AY=0$, and conversely any vector of the form $X0+Y$ is a solution.

Determine whether the given set S is a subspace of the Vector space V.

(there are multiple)

Refer to the system of linear equations $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$ Is the augmented matrix row-equivalent to its reduced row-echelon form?

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row echelon form. Solve the system. Assume that the variables are named $x_1,x_2,...$ form left to right. $\left[\begin{array}{ccccc}1& 0& 5& 3& 2\\ 0& 1& -2& 4& -7\\ 0& 0& 1& 1& 3\end{array}\right]$

The given matrix is the augmented matrix for a system of linear equations $[\begin{array}{}1& 0& -1& 0& -1& -2& 0\\ 0& 1& 2& 0& 1& 2& 0\\ 0& 0& 0& 1& 1& 1& 0\end{array}]$