Page 474 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(b) ,

(c) , ,

(d) , , ,

Let V be an inner product space. Show that if u and v are orthogonal unit vectors in V, then .
19.

Let V be an inner product space. Show that if w is orthogonal to both and , it is orthogonal to for all

20. scalars and . Interpret this result geometrically in the case where V is with the Euclidean inner product.

Let V be an inner product space. Show that if w is orthogonal to each of the vectors , , …, , then it is orthogonal to

21. every vector in span .

Let be a basis for an inner product space V. Show that the zero vector is the only vector in V that is
22. orthogonal to all of the basis vectors.

Let be a basis for a subspace W of V. Show that consists of all vectors in V that are orthogonal to
23. every basis vector.

Prove the following generalization of Theorem 6.2.4. If , , …, are pairwise orthogonal vectors in an inner product
24. space V, then

Prove the following parts of Theorem 6.2.2:
25.

(a) part (a)

(b) part (b)

(c) part (c)

Prove the following parts of Theorem 6.2.3:
26.

(a) part (a)

(b) part (b)

(c) part (c)