## Example 1

Find all real numbers $$x, y, z$$ satisfying the following system of linear equations: $\begin{array}{r} 3 x - 2y - z = 1 \\ 2z + x - y = 5 \end{array}$

## Example 2

Find all the solutions to the following system defined over $$GF(2)$$: \begin{eqnarray*} x_1 + x_2 + x_3 + x_4 & = & 1 \\ x_1 + x_2 + x_3 & = & 1 \\ x_2 + x_3 + x_4 & = & 0 \\ x_1 + x_2 + x_4 & = & 0 \end{eqnarray*}

## Example 3

Let $$A = \begin{bmatrix} 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix}$$. Let $$x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4\end{bmatrix}$$ be a tuple of variables. Find all solutions to $$A x = 0$$.

## Example 4

Let $$A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & -6 \\1 & 3 & 5\end{bmatrix}$$. Let $$B = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & -2 \\0 & 1 & 5\end{bmatrix}$$. Show that $$A$$ can be transformed into $$B$$ via a sequence of two elementary row operations.

## Example 5

Find all real numbers $$a$$ such that the system of linear equations \begin{eqnarray*} x - y + z & = & 1 \\ -x + 2y + z & = & -1 \\ y + az & = & 0 \end{eqnarray*} has infinitely many solutions. Justify your answer.

## Example 6

Consider the following system of linear equations defined over the field of real numbers: \begin{eqnarray*} x + y - 4z & = & 1 \\ -2x + y - z & = & -1 \\ x - y + 2z & = & 0. \end{eqnarray*} How many solutions to the system are there?

## Example 7

A system of linear equations is said to be inconsistent if it has no solutions. Otherwise it is said to be consistent. Is it possible for a system of linear equations defined over a field to be inconsistent if there are more variables than there are equations?

## Example 8

Let $$A$$ be an $$m\times n$$ matrix with entries from some field. Prove that no matter how you transform $$A$$ using elementary row operations to a matrix in RREF, you always end up with the same matrix.