Linear Algebra and Its Applications, Exercise 2.1.1

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 $(i, j)$ where $i$ and $j$ are integers; for example, $(0, 0)$ , $(1, 1)$ , $(-2, 3)$ , 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 $(1, 1)$ by $\frac{1}{2}$ produces a result $(\frac{1}{2}, \frac{1}{2})$ not in the set.

(b) The set of all points on the x axis and y axis, that is $(x_1, 0)$ and $(0, x_2)$ , 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.

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2 Responses to Linear Algebra and Its Applications, Exercise 2.1.1

zmok says:

July 8, 2013 at 3:03 am

a) the multiplication of two integer is always an integer,so the sum or difference of a integer and a fraction?, this is not closed under vector addition

- hecker says:
  
  July 8, 2013 at 3:09 pm
  
  I’m not sure exactly what you’re asking. There are two issues here: What types of numbers we use as entries in a vector, and what types of numbers we use as scalars to multiply vectors. By the definition of a vector space a scalar can potentially be any real number — you can’t restrict them to be integers, or fractions, or whatever. So you might construct a set of vectors using only integers as entries, but once you multiply those vectors by arbitrary scalars the entries in the resulting vectors could potentially be any real number.