Exercise 2.1.1. Construct the following:
(a) a subset of 2-D space closed under vector addition and subtraction but not scalar multiplication
(b) a subset of 2-D space closed under scalar multiplication but not vector addition
Answer: (a) The set of all vectors where and are integers; for example, , , , etc. This set is closed under vector addition, since the sum or difference of two integers is always an integer. However it is not closed under scalar multiplication, since (for example) multiplying by produces a result not in the set.
(b) The set of all points on the x axis and y axis, that is and , is closed under scalar multiplication but not under vector addition.
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.