Systems of Linear Equations and Matrix Methods

Linear systems describe intersections of lines or planes, resulting in three possible outcomes: a unique solution (one intersection point), infinitely many solutions (overlapping lines/planes), or no solution (parallel lines/planes).
The solution space is defined as the set containing all possible solutions to a system.
Systems are classified as consistent if at least one solution exists, and inconsistent if no solution can be found.

Gaussian elimination with back substitution involves converting an augmented matrix into row echelon form ( $ref$ ) using elementary row operations:
- Swapping two equations.
- Multiplying an equation by a non-zero scalar.
- Adding a scalar multiple of one equation to another.
Gauss-Jordan elimination extends this process to reach reduced row echelon form ( $rref$ ), where the first non-zero entry (pivot) in each row is $1$ and all entries above and below these pivots are $0$ .
The $rref$ of a matrix is unique, whereas the $ref$ is not.

Systems of equations are represented by a coefficient matrix ( $A$ ) and an augmented matrix ( $A|b$ ).
The rank ( $rank$ ) of a matrix is defined as the number of pivots in its echelon form.
A system is inconsistent if there is a pivot in the final column of its augmented matrix, which corresponds to an equation such as $0x + 0y = 1$ .
Consistent systems follow specific rank criteria based on the number of unknowns ( $n$ ):
- Unique solution: $rank(A) = rank(A|b) = n$ .
- Infinitely many solutions: $rank(A) = rank(A|b) < n$ .
Free variables occur when columns lack a pivot; these are expressed using parameters such as $s$ and $t$ to define a hyperplane.

Underdetermined systems have fewer equations than unknowns and typically result in infinitely many solutions or no solution; they cannot have a unique solution.
Overdetermined systems have more equations than unknowns and commonly have no solution, though unique or infinite solutions are possible.
Number of free variables is calculated as: $\text{total number of unknowns} - rank(A)$ .

Linear modeling can determine unit costs, such as the prices for figs, guavas, and honeydews purchased by Hewey ( $3, 2, 1$ for $8$ ), Dewey ( $1, 1, 1$ for $4$ ), and Louis ( $10, 2, 2$ for $16$ ).
Weight distributions can be solved by balancing conditions, such as determining weights for Bruce, Alex, Doris, and Chris given a total group weight of $500\,kg$ .
Systems can also be solved within the complex number plane, utilizing $i = \sqrt{-1}$ , for equations like $(1 + i)x - iy = -3$ and $2x - 2y = 3i$ .