How are the concepts of linear independence spanning and bas

How are the concepts of linear independence, spanning, and basis all related to each other?

Describe the answer in 3-4 sentences.

Solution

A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set .Linear Independence means we must be able to write the zero vector as a linear combination of the vectors given to us and the all constants must be equal to zero.

c1v1+ c2v2+c3v3----cnvn =0

c1=c2.....cn =0

A set of vectors S = {v1, v2, …, vn} are said to form a basis if the following conditions are met:

1. The set S spans all of their vector space.

2. The set S is linearly independent.

To show that a set of vectors for a basis, just show that they span all of their dimension OR show that they are linearly independent.

Spanning: It should be able to write every vector as a linear combination of the vectors given .