Exercise 3.1.19. State whether each of the following is true or false:
(a) If the subspaces and are orthogonal, then and are also orthogonal.
(b) If is orthogonal to and orthogonal to then is orthogonal to .
Answer: (a) In suppose that . Then since the zero vector is orthogonal to itself is orthogonal to . However we have and is not orthogonal to itself, so and are not orthogonal. The statement is false.
(b) Suppose that and . Then is orthogonal to and is orthogonal to but is not orthogonal to because is not orthogonal to itself. The statement is false.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.