Exercise 2.1.2. Of the following subsets of , which are subspaces and which are not?

(a) the set of vectors with the first component

(b) the set of vectors with the first component

(c) the set of vectors and for which

(d) the single vector

(e) all linear combinations of the vectors and

(f) all vectors for which

Answer: (a) This set is a subspace: It is closed under vector addition, since for two members of the set and the sum is also in the set. It is also closed under scalar multiplication, since for any vector the product is also in the set.

(b) This set is not a subspace: It is not closed under scalar multiplication, since for a vector the product is not in the set. It is also not closed under vector addition, since for the vectors and the sum is not in the set.

(c) This set is not a subspace: It is not closed under vector addition, since the vectors and are in the set but their sum is not.

(d) This set is a subspace: It is closed under vector addition, since the sum of and is . It is also closed under scalar multiplication, since for any scalar the product .

(e) This set is a subspace: It is closed under scalar multiplication, since for any vector the product is also a linear combination of and . It is also closed under vector addition, since for any two vectors and their sum is also a linear combination of and .

(f) This is a subspace: It is closed under scalar multiplication, since given a vector in the set, for the scalar product we have

It is also closed under vector addition, since given two vectors and in the set, for their sum we have

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.