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Home / Linear Algebra / Prove that, every basis of a vector space V has the same number of elements.

Prove that, every basis of a vector space V has the same number of elements.

Problem: Prove that, every basis of a vector space V has the same number of elements.

 Proof: Suppose {u1 , u2 , . . . , un } is a basis of  V, and suppose {v1 , v2 , . . .} is another basis of V. Because {ui} spans V, the basis {v1 ,v2, . . .} must contain n or less vectors, or else it is linearly dependent. On the other hand, if the basis        {v1,v2, . . .}  contains less than n elements, then {u1 , u2 , . . . , un } is linearly dependent. Thus, the basis fv {v1, v2, . . .} contains exactly n vectors, and so the theorem is true. ▄

Problem: Let V be a vector space of finite dimension n.     Then

  (i) Any n + 1 or more vectors must be linearly dependent.

 (ii) Any linearly independent set S = {u1 , u2 , . . . , un }  with n elements is a basis of  V.

(iii) Any spanning set T = {v1,v2, . ., vn}  of  V with n elements is a basis of V.

Proof: Suppose B = {w1,w2, . ., wn}   is a basis of  V.

  (i) Because B spans V, any n + 1 or more vectors are linearly dependent (ii), elements from B can be adjoined to S to form a spanning set of V with n elements.

  Because S already has n elements, S itself is a spanning set of V. Thus, S is a basis of  V.

(iii) Suppose T is linearly dependent. Then some v i is a linear combination of the preceding vectors. We know that V is spanned by the vectors in T without vi  and there are n –1 of them. We know that the independent set B cannot have more than n – 1 elements. This contradicts the fact that B has n elements. Thus, T is linearly independent, and hence T is a basis of V.▄

Problem:  Suppose S spans a vector space V. Then

 (i) Any maximum number of linearly independent vectors in S form a basis of V.

(ii) Suppose one deletes from S every vector that is a linear combination of preceding vectors in S. Then the remaining vectors form a basis of V.

Proof: (i) Suppose {v1,v2, . ., vm}  is a maximum linearly independent subset of S, and suppose w  ∊ S. Accordingly, {v1,v2, . ., vm} is linearly dependent. No vk can be a linear combination of preceding vectors.

 Hence, w is a linear combination of the vi . Thus, w ∊ span (vi ), and hence S   span (vi) . This leads to

                                       V= span(S) ⊆  span (vi ) ⊆ V

Thus, (vi) spans V, and, as it is linearly independent, it is a basis of V.

(ii) The remaining vectors form a maximum linearly independent subset of S; hence, by (i), it is a basis of V.▄

Problem: Let V be a vector space of finite dimension and let S = {u1 , u2 , . . . , ur }  be a set of linearly independent vectors in V. Then S is part of a basis of V ; that is, S may be extended to a basis of V.

Proof: Suppose B = {w1,w2, . ., wn} is a basis of V. Then B spans V, and hence V is spanned by

                                   S ⋃ B = {u1 , u2 , . . . , ur ,v1,v2, . ., vm}

We can delete from S⋃B each vector that is a linear combination of preceding vectors to obtain a basis Bˊ for V. Because S is linearly independent, no uk is a linear combination of preceding vectors. Thus, Bˊ contains every vector in S, and S is part of the basis Bˊ for V.▄

Problem: Let W be a subspace of an n-dimensional vector space V. Then dim

W ≤ n. In particular, if dim W= n, then W = V.

     Proof: Because V is of dimension n, any n +1 or more vectors are linearly dependent. Furthermore, because a basis of W consists of linearly independent vectors, it cannot contain more than n elements. Accordingly, dim W ≤ n.

     In particular, if {w1,w2, . ., wn} is a basis of W, then, because it is an independent set with n elements, it is

also a basis of V. Thus, W = V when dim W = n.▄

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