Proving the Validity of a Set of Vectors as a Basis for a Vector Space

[Mathman23]

Hi

Given a Vector Space V which has the basis $$\{ v_{1}, v_{2}, v_{3} \}$$ then I need to prove that the following set v = $$\{ v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$ is also a basis for V.

I know that in order for v to be a basis for V then $$V = span \{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$, and the vectors of v need to be linear independent.

but by showing that the vectors of v are linear independent is the best way of show that v is a basis for V?

Sincerely
Fred

 

[0rthodontist]

but by showing that the vectors of v are linear independent is the best way of show that v is a basis for V?

Probably.

Also do you have the theorem that if {b1, ... ,bn} is a basis for a vector space V, then any linearly independent set of n vectors in V is also a basis?

 

[benorin]

We know that every vector $$v\in V$$ can be expressed as a linear combination of the basis vectors $$\{ v_{1}, v_{2}, v_{3} \} ,$$ that is there exists scalars $$c_{1}, c_{2}, c_{3}$$ such that $$v=c_1v_1+c_2v_2+c_3v_3 .$$ Is the same true of $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$?

 

[Mathman23]

We know that every vector $$v\in V$$ can be expressed as a linear combination of the basis vectors $$\{ v_{1}, v_{2}, v_{3} \} ,$$ that is there exists scalars $$c_{1}, c_{2}, c_{3}$$ such that $$v=c_1v_1+c_2v_2+c_3v_3 .$$ Is the same true of $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$?

if I express $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$ as a set of vectors $$\{x,y,z\}$$ which can be expressed

$$v=c_1 x +c_2 y+c_3 z.$$

Then its the same.

Sincerely
Fred

 

[benorin]

I want to be sure you understand:

Suppose I take a linear combination of the vectors $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \} ,$$ this is then a linear combination of linear combinations of the given basis vectors, which is again, a linear combination of the given basis vectors, which every vector in V may be expressed as. Follow?

 

[Mathman23]

I get it ;)

Then I show that the linear combination of the linear combination er linear independent ??

like

c1 * v1 = 0

c2 * (v1+v2) = 0

c3 * (v1 + v2 + v3) = 0

Sincerly
Fred

I want to be sure you understand:

Suppose I take a linear combination of the vectors $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \} ,$$ this is then a linear combination of linear combinations of the given basis vectors, which is again, a linear combination of the given basis vectors, which every vector in V may be expressed as. Follow?

 

[benorin]

We know that every vector $$v\in V$$ can be expressed as a linear combination of the basis vectors $$\{ v_{1}, v_{2}, v_{3} \} ,$$ that is there exists scalars $$c_{1}, c_{2}, c_{3}$$ such that $$v=c_1v_1+c_2v_2+c_3v_3 .$$ Is the same true of $$\{v_{1}, v_{1}+ v_{2}, v_{1} + v_{2} + v_{3} \}$$?

Indeed, the existence of scalars $$c_{1}, c_{2}, c_{3}$$ such that $$v=c_1v_1+c_2v_2+c_3v_3$$ implies it, then there also exists scalars a,b,c such that $$v=av_1+b(v_1+v_2)+c(v_1+v_2+v_3).$$

Proof: Fix v. Let the above hypothesis be so. Then

$$v=av_1+b(v_1+v_2)+c(v_1+v_2+v_3)=(a+b+c)v_1 + (b+c)v_2+cv_3$$

but we know there exist scalars $$c_{1}, c_{2}, c_{3}$$ such that $$v=c_1v_1+c_2v_2+c_3v_3 ,$$

so set these equal to obtain

$$v=c_1v_1+c_2v_2+c_3v_3=(a+b+c)v_1 + (b+c)v_2+cv_3$$

equate coefficients of the v_k's to get the systems of equations $$c_1=a+b+c,c_2=b+c,c_3=c$$ so that we may take $$c=c_3, b=c_2-c_3,a=c_1-c_2$$ and hence there must also exists scalars a,b,c such that $$v=av_1+b(v_1+v_2)+c(v_1+v_2+v_3),$$ as required.

 

[Mathman23]

Here is my own idear for a proof that the set of vectors $$v = \{v_{1}, v_{1} + v_{2},v_{1} + v_{2} + v_{3} \}$$ is a basis for the vector space V.

Definition: Basis for Vector Space

Let V be a Vector Space. A set of vectors in V is a basis for V if the following conditions are meet:

1. The set vectors spans V.
2. The set of all vectors is linear independent.

Proof:

(2) The vectors in v is said to be linear independent iff there doesn't exists scalars $$C = (c_1,c_2,c_3) \neq 0$$ such that

$$v = c_1 (v_1) + c_2 (v_1+v_2) + c_3 (v_1 + v_2 + v_3) = 0$$

By expression the above in matrix-equation form:

$\begin{array}{cc}\[ \left[ \begin{array}{ccc} <br /> v_{11}& (v_{11} + v_{12})& (v_{11} + v_{12} + v_{13}) \\<br /> v_{21}& (v_{21} + v_{22})& (v_{21} + v_{22} + v_{23})\\<br /> v_{31}& (v_{31} + v_{32})& (v_{31} + v_{32} + v_{33})\\ <br /> \end{array} \right]\] \cdot \left[ \begin{array}{c} c_1 \\ c_2 \\ c_3 \end{array} \right] = 0 \] \end{array}$

If v is fixed, it can be concluded that matrix columbs of v are linear independent cause the only solution for the equation vC = 0 is the trivial solution(zero-vector):

$\begin{array}{cc}\[ \left[ \begin{array}{ccc} <br /> c_{1} \\<br /> c_{2} \\<br /> c_{3} \\ <br /> \end{array} \right]\] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right] \] \end{array}$

Which implies the dependence relation between v and C doesn't exist since C = 0.

(1) Since the only solution for matrix equation is the trivial solution, then according to the definition for the Span of a vector subspace, then the set of v spans V.

This proves that the set of is it fact a basis for the Vector Space V.

Is my proof valid?

Sincerely
Fred