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Algebra I: A Basic Course in Abstract Algebra by Rajendra Kumar Sharma, Sudesh Kumari Shah, Asha Gauri Shankar

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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 ...

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