Page 467 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

which proves that and are in .

Proof (b) If v is common to W and , then , which implies that by Axiom 4 for inner products.

Remark Because W and are orthogonal complements of one another by part (c) of the preceding theorem, we shall say
that W and are orthogonal complements.

A Geometric Link between Nullspace and Row Space

The following fundamental theorem provides a geometric link between the nullspace and row space of a matrix.

THEOREM 6.2.6

If A is an matrix, then

(a) The nullspace of A and the row space of A are orthogonal complements in with respect to the Euclidean inner
product.

(b) The nullspace of and the column space of A are orthogonal complements in with respect to the Euclidean
inner product.

Proof (a) We want to show that the orthogonal complement of the row space of A is the nullspace of A. To do this, we must

show that if a vector v is orthogonal to every vector in the row space, then , and conversely, that if , then v is

orthogonal to every vector in the row space.

Assume first that v is orthogonal to every vector in the row space of A. Then in particular, v is orthogonal to the row vectors ,
, …, of A; that is,

(9)

But by Formula 11 of Section 4.1, the linear system can be expressed in dot product notation as

(10)

so it follows from 9 that v is a solution of this system and hence lies in the nullspace of A.

Conversely, assume that v is a vector in the nullspace of A, so . It follows from 10 that

But if r is any vector in the row space of A, then r is expressible as a linear combination of the row vectors of A, say