Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V ,

what is the relationship between S and the basis of V ?

Expert Solution

The relationship between S and the basis of V is either S or proper subset of S is a basis of V.There exist a basis for every finitely generated vector space V.The proof is given below.

Proof. Case 1. Let V be a finitely generated vector space other than
the null space.
Let S = {a1, a2, ...,an} be a finite set of generators of V. If S is
linearly independent in V, then S itself is a basis of V and the theorem
is done.
If S is linearly dependent, then we can delete, by Deletion theorem,
at least one vector from S and obtain a proper subset S1 (of S) spanning
the same space V. If S1 is linearly independent in V, then S1 is a basis
of V and the theorem is done.
If S, be not linearly independent then we repeat the process of dele-
tion and finally obtain, after k(<n) steps of deletion, a subset Sk which
is linearly independent in V and which also spans V.
This is possible, because S is a finite set of n elements and in the
extreme unfavourable case we can come, after n - 1 steps of deletion, to
a subset Sn-1 containing only one non-zero vector that generates V and
that is linearly independent.
Therefore our assertion that S is a linearly independent set for some
k(<n) is true and hence it is a basis of V.
Case 2. Let V = {0}, 0 is the null vector of V. Since the null(/void set which is denoted by Φ) set is linearly independent and
L(Φ) = {0}, Φ is a basis of V.
This completes the proof.

Colby Messinger answered 2 years ago

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