Page 256 - A Course in Linear Algebra with Applications

2 4 0 Chapter Seven: Orthogonality in Vector Spaces

3
5. Find the projection of the vector ( 4 | on the subspace
-2_
of R 3 which has the orthonormal basis consisting of

6. Express the matrix of Exercise 3 in QR-factorized form.
7. Show that a non-singular complex matrix can be expressed
as the product of a unitary matrix and an upper triangular
matrix whose diagonal elements are real and positive.

8. Find a factorization of the type described in the previous
exercise for the matrix

( — i i
\l + i 2

where % = ->/—l.
-1
9. If A and B are orthogonal matrices, show that A and AB
are also orthogonal. Deduce that the set of all real orthogonal
nxn matrices is a group with respect to matrix multiplication
in the sense of 1.3.
10. If A = QR — Q'R' are two QR-factorizations of the real
non-singular square matrix A, what can you say about the
relationship between the Q and Q', and R and R'l
11. Let L be a linear operator on the Euclidean inner product
n
space R . Call L orthogonal if it preserves lengths, that is, if
n
||LpO|| = \\X\\ for all vectors X in R .
(a) Give some natural examples of orthogonal linear
operators.
(b) Show that L is orthogonal if and only if it preserves
inner products, that is, < L(X),L(Y) > = < X,Y >
for all X and Y.