# Basis

A set of vectors u1,u2,…,un is said to be linearly dependent if there are k1,k2,…,knЄR, such that:

k1u1 + k2u2 +….+ knun= 0        with some ki ≠ 0

That is, one of them is a linear combination of the others. Otherwise, the set is said to be linearly independent.

A set of vectors is said to be a generative system if all vectors can be expressed as linear combination of these vectors.

A basis is a set of vectors linearly independent and generative system.

In R3 , B = {u1, u2, u3} is a basis if:

–They are linearly independent

–It is a generative system, that is, each vector v in R3 can be expressed as v = λ1u1 + λ2u2 + λ3u3. The (λ1, λ2, λ3) are the coordinates of v with respect to B.

Basis Theorem (or dimension Theorem): every basis in a vector space has the same number of elements. This number is called dimension of the vector space.

In R3 ,  we have the canonical basis:

Then, if A(a1,a2,a3) and B(b1,b2,b3)

and there always exists an equipollent vector

To calculate the magnitude, we use Pythagorean Theorem:

Then, it is easy to check that:

If |v| = 1, it is called unit vector.

The operations in coordinates are:

Exercises:

1.- If u(3,-2,1), v(1,3,-2), A(1,t,2), B(-3,3,0):

a) Calculate the magnitude of u.

b) Calculate t, if AB(-4,-2,-2).

c) Calculate u + v, 2u - 5v

2.- Decide if the following sets of vectors are linearly dependent, linearly independent, generative system and/or basis:

a) B1={(1,0,0),(1,1,0),(1,1,1)}

b) B2={(1,0,0),(0,1,0),(0,0,1),(1,1,1)}

c) B3={(1,1,1),(1,-1,3),(-1,3,-5)}

Solutions:

1.- a)|u|= √14; b) 5; c) u + v = (4,1,-1); 2u - 5v = (1,-19,12)

2.- a) Basis, generative system and linearly independent; b) linearly dependent and generative system; c) linearly dependent