Linear Algebra and Its Applications · Problem 3.1.43

Orthogonality — Gilbert Strang

Mathematics · orthogonal-subspaces

Problem

If PP is the plane of vectors in R4\mathbf{R}^4 satisfying x1+x2+x3+x4=0x_1 + x_2 + x_3 + x_4 = 0, write a basis for P⊥P^\perp. Construct a matrix that has PP as its nullspace.
Intuitive explanation

The key insight is that P⊥P^\perp consists of all vectors orthogonal to every vector in PP—and since PP is defined by a single linear constraint, P⊥P^\perp is spanned by the normal vector to that constraint. Once you identify the normal, you can build a matrix whose rows are orthogonal to PP, making PP exactly its nullspace.

Intuitive explanation

The key insight is that P⊥P^\perp consists of all vectors orthogonal to every vector in PP—and since PP is defined by a single linear constraint, P⊥P^\perp is spanned by the normal vector to that constraint. Once you identify the normal, you can build a matrix whose rows are orthogonal to PP, making PP exactly its nullspace.

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