Page 224 - A Course in Linear Algebra with Applications

2L)o Chapter Seven: Orthogonality in Vector Spaces

3
7. If X, Y, Z are vectors in R , prove that

T
T
T
X (Y x Z) = Y (Z xX) = Z (X x Y).
(This is called the scalar triple product of X, Y, Z). Then
show that that the absolute value of this number equals the
volume of the parallelopiped formed by line segments repre-
senting the vectors X, Y, Z drawn from the same initial point.
8. Use Exercise 7 to find the condition for the three vectors
X, Y, Z to be represented by coplanar line segments.
9. Show that the set of all vectors in R n which are orthog-
n
onal to a given vector X is a subpace of R . What will its
dimension be?
n
10. Prove the Cauchy-Schwartz Inequality for R . [Hint:
2
T
compute the expression ||X|| ||y|| 2 — X F| 2 and show that
|
it is is non-negative].
11. Find the most general vector in C 3 which is orthogonal
to both of the vectors

( -s*\ ( x \

2 + 7=T and 1 .
V 3 ) \J=2)

12. Let A and B be complex matrices of appropriate sizes.
Prove the following statements:

(a)(i) T = (W); (b)(A + B)* = A*+B*; (c)(A*)* = A.

13. How should the vector projection of X on Y be defined
3
inC ?