# Basis

*u*is said to be

_{1},u_{2},…,u_{n}**linearly dependent**if there are k

_{1},k

_{2},…,k

_{n}Є

**R**, such that:

k_{1}*u*_{1} + k_{2}*u*_{2} +….+ k_{n}*u*_{n}= 0 with some k_{i} ≠ 0

**linearly independent**.

**generative system**if all vectors can be expressed as linear combination of these vectors.

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

**R**

^{3},

*B = {u*is a basis if:

_{1}, u_{2}, u_{3}}*v*in

**R**

^{3}can be expressed as v = λ

_{1}u

_{1}+ λ

_{2}u

_{2}+ λ

_{3}u

_{3}. The (λ

_{1}, λ

_{2}, λ

_{3}) 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**.

**R**

^{3}, we have the canonical basis:

_{1},a

_{2},a

_{3}) and B(b

_{1},b

_{2},b

_{3})

and there always exists an equipollent vector

*v*| = 1, it is called

**unit vector**.

**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) B_{1}={(1,0,0),(1,1,0),(1,1,1)}

b) B_{2}={(1,0,0),(0,1,0),(0,0,1),(1,1,1)}

c) B_{3}={(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

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