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P. 837
Chapter 10
Supplementary Exercises
Let and be vectors in , and let and .
1. .
(a) Prove:
(b) Prove: and are orthogonal if and only if and are orthogonal.
Show that if the matrix
2.
is nonzero, then it is invertible.
Find a basis for the solution space of the system
3.
Prove: If and are complex numbers such that , and if is a real number, then
4.
is a unitary matrix.
Find the eigenvalues of the matrix
5.
where .
Consider the relation between the complex number and the corresponding matrix with real entries
6.
.
(a) How are the eigenvalues of related to ?
