Online Test Banks
Score higher
See Online Test Banks
Learning anything is easy
Browse Online Courses
Mobile Apps
Learning on the go
Explore Mobile Apps
Dummies Store
Shop for books and more
Start Shopping

How to Meet Vector Space Requirements

Part of the Linear Algebra For Dummies Cheat Sheet

In linear algebra, a set of elements is termed a vector space when particular requirements are met. For example, let a set consist of vectors u, v, and w. Also let k and l be real numbers, and consider the defined operations of ⊕ and ⊗. The set is a vector space if, under the operation of ⊕, it meets the following requirements:

  • Closure. uv is in the set.

  • Commutativity. uv = vu.

  • Associativity. u ⊕ (vw) = (uv) ⊕ w.

  • An identity element 0. u0 = 0u = u for any element u.

  • An inverse element −u. u−u = −u u = 0

Under the operation of ⊗, the set is a vector space if it meets the following requirements:

  • Closure. ku is in the set.

  • Distribution over a vector sum. k ⊗ (uv) = kukv.

  • Distribution over a scalar sum. (k + l) ⊗ u = kulu.

  • Associativity of a scalar product. k ⊗ (lu) = (kl) ⊗ u.

  • Multiplication by the scalar identity. 1 ⊗ u = u.

blog comments powered by Disqus

Inside Sweepstakes

Win an iPad Mini. Enter to win now!