Activity 2.2.4

Consider the linear system $$\begin{alignedat}{4} 2x & {}+{} & y & {}-{} & 3z & {}={} & 4 \\ -x & {}+{} & 2y & {}+{} & z & {}={} & 3 \\ 3x & {}-{} & y & & & {}={} & -4. \\ \end{alignedat}$$ Identify the matrix $A$ and vector $\bvec$ to express this system in the form $A\xvec = \bvec$.

If $A$ and $\bvec$ are as below, write the linear system corresponding to the equation $A\xvec=\bvec$ and describe its solution space, using a parametric description if appropriate: $$A = \left[\begin{array}{rrr} 3 & -1 & 0 \\ -2 & 0 & 6 \end{array} \right],~~~ \bvec = \left[\begin{array}{r} -6 \\ 2 \end{array} \right].$$

Describe the solution space of the equation $$\left[ \begin{array}{rrrr} 1 & 2 & 0 & -1 \\ 2 & 4 & -3 & -2 \\ -1 & -2 & 6 & 1 \\ \end{array} \right] \xvec = \left[\begin{array}{r} -1 \\ 1 \\ 5 \end{array} \right]$$.

Suppose $A$ is an $m\times n$ matrix. What can you guarantee about the solution space of the equation $A\xvec = \zerovec$?