Linear Systems with Parameters

Definition Let $S$ be a set of real numbers. A family of linear systems assigns to each $a$ in $S$ a linear system $A(a)\mathbf{x} = \mathbf{b}(a)$, where the entries of the $m \times n$ matrix $A(a)$ and of the vector $\mathbf{b}(a)$ are given expressions in $a$, defined for every $a$ in $S$. A letter such as $a$ that appears among the coefficients is a coefficient parameter. It is not an unknown. For each fixed value of $a$ the coefficients are ordinary numbers, and the system is an ordinary linear system in the unknowns with its own solution set. Unless a problem restricts it, $S$ contains every real number at which these expressions are defined, negative values included.

Definition Multiplying an equation by an expression $c(a)$, or dividing it by $c(a)$, gives an equivalent equation for every value of $a$ with $c(a) \ne 0$. When $c(a) = 0$, division is undefined, and multiplication replaces the equation by $0 = 0$ and loses its information. So a step that divides by $c(a)$, scales a row by $c(a)$, or cancels a common factor $c(a)$ from both sides is valid only under the condition $c(a) \ne 0$, and every result derived after that step carries the same condition. A row replacement $R_i \to R_i + c(a)R_j$ with $i \ne j$ needs no condition when the multiplier $c(a)$ is defined for every value of $a$, as a polynomial in $a$ is, because $R_i \to R_i - c(a)R_j$ then undoes it for every value of $a$. A multiplier such as $1/(a - 1)$ hides a division and is defined only for $a \ne 1$.

Definition In a family solved by elimination, an exceptional parameter value is a value at which an expression used as a divisor in deriving the general solution equals zero, even if the final formula shows no denominator. Usually that expression is the entry that serves as a pivot for all other values. At an exceptional value the general formula was never derived, so it does not apply, and the system has to be solved again from the equations obtained before the division or from the original equations. If the row containing that divisor has no other nonzero coefficient, the row reads $0 = c$ at the exceptional value, for some number $c$. When $c \ne 0$ the system has no solution. When $c = 0$ the row places no condition, and the remaining rows decide the solution set.

Definition Case analysis splits the allowed parameter values into cases that together cover every value, chosen so that each step used within a case is valid for every value in that case. Typically one case holds every value at which all divisors are nonzero, and the other cases cover the values at which some divisor is zero. Each case is solved separately. The result is a classification of the family, which states for every parameter value whether the system has exactly one solution, no solution or infinitely many, and gives all the solutions.

Definition A coefficient parameter is fixed before the system is solved. Different values can give different systems, and the answer is stated for each value. A free variable belongs to one fixed, consistent system. It is an unknown whose column has no pivot in an echelon form, it may take any real value $t$, and once values are chosen for all the free variables the pivot variables are determined. Changing $t$ moves between solutions of the same system without changing any equation. A letter in a proposed solution formula is free only if the formula satisfies all the equations for every value of that letter.