Question

Explain why you need at least ‘m’ vectors to span a space of dimension ‘m’. See Theorem 3.3.4b.

Short answer

We need at least ‘m’ vectors to span a space of dimension ‘m’.

Step-by-step solution

To prove the given statement

Let V be a vector space of dimension ‘m’

Let us consider a set G consisting of ‘m’ vectors such that G is a finite generating set for V. Then by theorem, “If a vector space V is generated by a finite set S , then some subset of S is a basis for V and hence, V has a finite basis”, some subset H of G is a basis for V. Now by theorem, “If V be a vector space having a finite basis, then every basis for V contains the same number of vectors”. This implies that the subset H of G contains exactly ‘m’ vectors.

Since a subset H of G contains ‘m’ vectors then G must contains at least ‘m’ vectors.

   Final Answer

Since the generating set G of vector space contains at least ‘m’ vectors, therefore, we need at least ‘m’ vectors to span a space of dimension ‘m’.