Page 242 - A Course in Linear Algebra with Applications

226 Chapter Seven: Orthogonality in Vector Spaces

7. Let A be a real n x n matrix with properties (a), (b) and
(c) of Exercise 6. Prove that < X, Y > = X T AY defines an
n T
inner product on R . Deduce that ||X|| = y/X AX defines a
n
norm on R .
8. Let S be the subspace of the inner product space -Ps(R)
2 2
generated by the polynomials 1 — x and 2 — x + x , where
< /, g > = f Q f(x)g(x)dx. Find a basis for the orthogonal
complement of S.
9. Find the projection of the vector with entries 1, —2, 3 on
/ l 0
the column space of the matrix 2 —4

\ 3 5
10. Prove the following statements about subspaces S and T
of a finite dimensional real inner product space:
±
±
(a) (S + T) 1 =S DT ;
1
1
(b) S - = T - always implies that S = T;
1 ± ±
(c) (SDT) - = S +T .
11. If S is a subspace of a finite dimensional real inner product
1
space V, prove that S - ~ V/S.

7.3 Orthonormal Sets and the Gram-Schmidt Process

Let V be an inner product space. A set of vectors in V is
called orthogonal if every pair of distinct vectors in the set is
orthogonal. If in addition each vector in the set is a unit vec-
tor, that is, has norm is 1, then the set is called orthonormal.

Example 7.3.1
3
In the Euclidean space R the vectors