Linear Algebra and Its Applications · Problem 3.1.49

Orthogonality — Gilbert Strang

Mathematics · perpendicular-columns-rows

Problem

Why is each of these statements false? ((a))

(1,1,1)

is perpendicular to

(1,1,-2)

, so the planes

x+y+z=0

and

x+y-2z=0

are orthogonal subspaces. ((b)) The subspace spanned by

(1,1,0,0,0)

and

(0,0,0,1,1)

is the orthogonal complement of the subspace spanned by

(1,-1,0,0,0)

and

(2,-2,3,4,-4)

. ((c)) Two subspaces that meet only in the zero vector are orthogonal.