Parametric Solutions

Linear Algebra (Matrices) · Axiom Academy

When a system has infinitely many solutions, set the free variables to parameters and sweep out the whole solution set as a point plus a span. 1. Free Variables Come From the RREF Take the system below. Row-reduce it and the second row collapses — it was just the first. One independent equation, three unknowns. Augmented matrix, then its reduced row echelon form In the RREF, the column with a leading 1 is a pivot column — its variable is determined. Columns with no pivot are free — those variables can be anything. 2. A Free Variable Becomes a Slider Watch one free variable in action. Hold x_2 = 0 and let x_3 = t be the slider. The lone equation x_1 + x_3 = 3 forces the pivot x_1 = 3 - t , so the solution is . As t slides, the pivot x_1 updates and the solution point sweeps along a line . Every swept point is a real solution At t=1 the point is (2,0,1) ; check: 2 + 2(0) + 1 = 3 . ✓ At t=-2 it is (5,0,-2) ; check: 5 + 2(0) - 2 = 3 . ✓ The slider never leaves the solution set — that's the whole point of a parameter. 3. Particular Solution + a Span With both free variables turned on ( x_2 = s , x_3 = t ), solving the pivot gives x_1 = 3 - 2s - t . Split that into a fixed particular solution plus a span of two direction vectors — one carried by each parameter. Particular solution : set every parameter to 0 . It anchors the set — one guaranteed solution to build from. Direction vectors and : each lies in the null space ( ), so adding any amount of them keeps you on solutions.

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