What is system? — Mathematics

Definition 22.13 University Mathematics — Year 1 · Chapter 22 — Determinants and Linear Systems

A linear system of $n$ equations in $p$ unknowns is $AX = B$ with $A \in \mathcal{M}_{n,p}(K)$ , $B \in K^n$ ; it is homogeneous when $B = 0$ . Its solution set, when nonempty, is $X_0 + \ker A$ : a particular solution plus the general homogeneous solution — an affine subspace of dimension $p - \operatorname{rk} A$ (rank–nullity).

Examples

Example 22.14 (The affine structure, made visible)

Solve

\begin{cases} x + y + z = 3\\ x - y + 2z = 2 . \end{cases}

Subtracting the equations: $2y - z = 1$ , so $z = 2y - 1$ and $x = 3 - y - z = 4 - 3y$ . The solutions form the line

(x, y, z) = (4,\ 0,\ -1) + y\,(-3,\ 1,\ 2) \qquad (y \in \R):

the particular solution $X_0 = (4, 0, -1)$ (the choice $y = 0$ ) plus the kernel line $\ker A = \operatorname{Vect}(-3, 1, 2)$ of the associated homogeneous system — check: $(-3) + 1 + 2 = 0$ and $(-3) - 1 + 4 = 0$ . Geometrically, two non-parallel planes of $\R^3$ intersect along a line, and the dimension count $p - \operatorname{rk} A = 3 - 2 = 1$ knew it before we solved anything. Changing the particular solution (say $y = 1$ : $X_0' = (1, 1, 1)$ ) changes the description, not the line: an affine subspace has many origins and one direction.

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Examples

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