In the homogeneous system Ax = 0, each row rᵢ imposes the condition rᵢ · x = 0. In ℝ² this defines a line through the origin, and in ℝ³ it defines a plane through the origin. The null space of A is the common intersection of all these row-normal subspaces.
The row view turns matrix equations into geometric constraints.
The chapter ends by distinguishing subspaces, complementary subspaces and orthogonal subspaces—three ideas needed to describe how row constraints divide a vector space.
A vector x belongs to null(A) exactly when every row of A maps it to zero. Therefore x must satisfy every equation rᵢ · x = 0 at the same time. Geometrically, it must lie in every row-normal line or plane, so null(A) is their common intersection.
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| Topic | Pages |
|---|---|
| Row view of a matrix | 1 |
| Images of row vs column views | 2–5 |
| 3D images of rows | 6 |
| Summary of definitions | 6–7 |
Independent rows impose independent constraints. In a full-rank 2×2 matrix, two distinct row-normal lines leave only the origin in their intersection. When the rows are dependent, the constraints repeat the same geometric condition, so the intersection remains a line. The same idea extends to three row planes in ℝ³.