## Chapter 14

## Basis and Dimension

In the previous chapters we have seen that given a vector space *V* for example *R*^{n}, 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 ℝ^{3} over ℝ, consider

*v*_{1} = (1,1,0), *v*_{2} = (1, 0, 1), *v*_{3} = (2, 1, 1), *v*_{4} = (0, 1, 1)

Let *S = {v*_{1}, v_{2}, v_{3}, v_{4}}. Then *S* spans ℝ^{3}. We see that *v*_{3} = *v*_{1} + *v*_{2}, i.e.

*v*_{3} = *v*_{1} + *v*_{2} + 0.v_{4} (14.1)

Hence *v*_{3} is a linear combination of *v*_{1},v_{2} and *v*_{4}, so that any linear combination of ...