Question

What is called a basis for a vector space? What are the extra properties you expect for a good basis?why?

Answer 1

basis for a vector space: Let V be a subspace of Rⁿ for some n. A collection B = {V₁,V ₂, …,V_r } of vectors from V is said to be a Basis for V if B is linearly independent and spans V. If either one of these criterial is not satisfied, then the collection is not a basis for V.

If a collection of vectors spans V, then it contains enough vectors so that every vector in V can be written as a linear combination of those in the collection.

If the collection is linearly independent, then it doesn't contain so many vectors that some become dependent on the others.

then a basis has just the right size: It's big enough to span the space but not so big as to be dependent.

example:  The collection { i, i+j, 2 j} is not a basis for R ². Although it spans R ², it is not linearly independent. No collection of 3 or more vectors from R ² can be independent

The properties of the basis of a vector space are given below,Let us suppose that B be a basis of vector space V

1) The set B is the smallest generating set for V. We can say that there is no proper subset of B which is generating set of V.

2)The set B is the largest set of linearly independent vectors. Thus, there is no another linearly independent set which contains B as a proper subset.