S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 2: Rings, Fields and Vector Spaces / From my notebooks

N19 Span and Linear Independence

In progress: these notes stop partway.

Linear combinations

A vector is a linear combination of if it can be written as for some . But if there are no (no solutions), then it is not a linear combination.

Example N19.1. is a linear combination of and in because :

Exercise N19.2 (Axler 2A.12). Suppose is linearly independent in and . Prove that if is linearly dependent, then .

Show solution
  • Do know: linear independence [of ] in .
  • Don’t know: .
  • WTS: is linearly dependent .