- #1

- 96

- 0

If W is subset of a vector space V, then W is a subspace of V if and only if the following TWO conditions hold

1) If u and v are vectors in W, then u+v is in W

2) If k is any scalar and u is any vector in W, then ku is in W

However, Lang, S. Introduction to linear algebra (UTM, page 91) adds another condition besides the above two mentioned conditions, which is

3) The element O of V is also an element of W.

wiki has the same treatment as Lang' book. It seems that 3) can be obtained from 2) by setting k=0. The question is whether 3) is necessary? Is there anything I miss in understanding the books and wiki?