Algebra I: A Basic Course in Abstract Algebra

Chapter 14

Basis and Dimension

In the previous chapters we have seen that given a vector space V for example Rⁿ, we can find a subset S of V such that S spans V. In this chapter we are interested in finding a subset S of V which spans V and no proper subset of S can span V. Such a set is called a minimal spanning set. We will show that S is such a set if no element of S is a linear combinations of the remaining elements.

14.1 Linearly Dependent Sets

In the vector space ℝ³ over ℝ, consider

 

v₁ = (1,1,0), v₂ = (1, 0, 1), v₃ = (2, 1, 1), v₄ = (0, 1, 1)

 

Let S = {v₁, v₂, v₃, v₄}. Then S spans ℝ³. We see that v₃ = v₁ + v₂, i.e.

 

v₃ = v₁ + v₂ + 0.v₄       (14.1)

 

Hence v₃ is a linear combination of v₁,v₂ and v₄, so that any linear combination of ...