Abstract Algebra Theory and Applications - Computer Science ...

18.3 LINEAR INDEPENDENCE 317Proposition 18.5 Suppose that a vector space V is spanned by n vectors.If m > n, then any set of m vectors in V must be linearly dependent.A set {e 1 , e 2 , . . . , e n } of vectors in a vector space V is called a basis forV if {e 1 , e 2 , . . . , e n } is a linearly independent set that spans V .Example 7. The vectors e 1 = (1, 0, 0), e 2 = (0, 1, 0), and e 3 = (0, 0, 1)form a basis for R 3 . The set certainly spans R 3 , since any arbitrary vector(x 1 , x 2 , x 3 ) in R 3 can be written as x 1 e 1 + x 2 e 2 + x 3 e 3 . Also, none of thevectors e 1 , e 2 , e 3 can be written as a linear combination of the other two;hence, they are linearly independent. The vectors e 1 , e 2 , e 3 are not the onlybasis of R 3 : the set {(3, 2, 1), (3, 2, 0), (1, 1, 1)} is also a basis for R 3 . Example 8. Let Q( √ 2 ) = {a + b √ 2 : a, b ∈ Q}. The sets {1, √ 2 } and{1 + √ 2, 1 − √ 2 } are both bases of Q( √ 2 ). From the last two examples it should be clear that a given vector spacehas several bases. In fact, there are an infinite number of bases for bothof these examples. In general, there is no unique basis for a vector space.However, every basis of R 3 consists of exactly three vectors, and every basisof Q( √ 2 ) consists of exactly two vectors. This is a consequence of the nextproposition.Proposition 18.6 Let {e 1 , e 2 , . . . , e m } and {f 1 , f 2 , . . . , f n } be two bases fora vector space V . Then m = n.Proof. Since {e 1 , e 2 , . . . , e m } is a basis, it is a linearly independent set. ByProposition 18.5, n ≤ m. Similarly, {f 1 , f 2 , . . . , f n } is a linearly independentset, and the last proposition implies that m ≤ n. Consequently, m = n.□If {e 1 , e 2 , . . . , e n } is a basis for a vector space V , then we say that thedimension of V is n and we write dim V = n. We will leave the proof ofthe following theorem as an exercise.Theorem 18.7 Let V be a vector space of dimension n.1. If S = {v 1 , . . . , v n } is a set of linearly independent vectors for V , thenS is a basis for V .2. If S = {v 1 , . . . , v n } spans V , then S is a basis for V .