How is a system of linear equations written as a matrix equation of the form \(A\mathbf{x} = \mathbf{b}\)?
\(A\) is the coefficient matrix containing the coefficients of each variable, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the column vector of constant terms from the right side of each equation.
What is an augmented matrix for a linear system?
An augmented matrix appends the column vector of constants \(\mathbf{b}\) to the coefficient matrix \(A\), typically separated by a vertical bar: \([A \mid \mathbf{b}]\). It contains all essential numerical data needed to solve the system.
What are the three allowable Elementary Row Operations (EROs) in Gaussian elimination?
1. Swap any two rows (\(R_i \leftrightarrow R_j\)).
2. Multiply all entries in a row by a non-zero scalar (\(k R_i \to R_i\), where \(k \neq 0\)).
3. Add a scalar multiple of one row to another row (\(R_i + k R_j \to R_i\)).
What is the key difference between Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)?
In REF, the leading entry (pivot) in each non-zero row is \(1\), with all entries below each pivot equal to \(0\). In RREF, each pivot is \(1\) and it is the ONLY non-zero entry in its entire column (zeros both above and below each pivot).
What row pattern in an augmented matrix signals that a system has no solution (is inconsistent)?
A row of the form \([0 \quad 0 \quad \dots \quad 0 \mid k]\) where \(k \neq 0\). This represents the false mathematical statement \(0 = k\).
What row pattern in an augmented matrix indicates that a consistent system has infinitely many solutions?
A row entirely of zeros, \([0 \quad 0 \quad \dots \quad 0 \mid 0]\), provided there are fewer non-zero rows than variables and no contradictory rows exist. Variables without pivot columns become free parameters.
Solve the system using row operations on an augmented matrix: $$\begin{cases} x + 2y = 7 \\ 3x - y = 7 \end{cases}$$
What is a 'free variable' in the context of solving a dependent linear system?
A variable that does not correspond to a pivot column in row echelon form. It can take on any arbitrary real value \(t\), and all other pivot variables are expressed in terms of this parameter.
