Ordered basis and linear independence

[Susanne217]

Homework Statement

I have a set of Vector v_1,v_2,v_3,v_4 in \mathbb{R}^4 and need to show that E = v_1,v_2,v_3,v_4 is an ordered basis for \mathbb{R}^4

The Attempt at a Solution

I know that for this being the case

v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 where v \in \mathbb{R}^4 but if the vectors are linear independent if there doesn't exist any set of unique scalars that makes the linear combination above true other than c_1 = c_2 = c_3 = c_4 = 0 and thus v = {0}. Then these vectors are linear independent and an ordered basis for \mathbb{R}^4. right?

Best regards
Susanne

 

[owlpride]

Four vectors v1, v2, v3, v4 are a basis for R^4 if the equation

<br /> 0 = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 <br />

has the unique solution <br /> c_1 = c_2 = c_3 = c_4 = 0<br />.

More generally, if v1, v2, v3, v4 are a basis, then the equation
<br /> v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 <br />
has a unique solution for any v, but you only need to check it for v=0 to make sure that you have a basis.

 

[Susanne217]

Four vectors v1, v2, v3, v4 are a basis for R^4 if the equation

<br /> 0 = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 <br />

has the unique solution <br /> c_1 = c_2 = c_3 = c_4 = 0<br />.

More generally, if v1, v2, v3, v4 are a basis, then the equation
<br /> v = c_1 \cdot v_1 + \cdots + c_4\cdot v_4 <br />
has a unique solution for any v, but you only need to check it for v=0 to make sure that you have a basis.

so what you are saying owlpride is that if I can show that the linear-combination has the where they are linear independent then they form not just a basis but an ordered basis for R4?

 

[owlpride]

The difference between an ordered basis and a basis is just an explicit order of the basis vectors. For example, the set {(1,1), (0,1)} is a basis for R^2. It becomes an ordered basis when I declare that v1 = (1,1) and v2 = (0,1). Once you have an ordered basis, you can work with coordinates. For example, relative to this ordered basis the coordinates (1,3) refer to the vector 1*v1 + 3*v2 = (1,4) (relative to the standard basis (1,0) and (0,1)).

You cannot work with coordinates until you specify an order on the basis vectors. Without an order, you would not be able to tell if the coordinates (1,3) refer to 1*(1,1) + 3*(0,1) or 1*(0,1) + 3*(1,1).

Don't get caught up on the word "ordered". All you need to prove is that your vectors form a basis.

 

[Susanne217]

The difference between an ordered basis and a basis is just an explicit order of the basis vectors. For example, the set {(1,1), (0,1)} is a basis for R^2. It becomes an ordered basis when I declare that v1 = (1,1) and v2 = (0,1). Once you have an ordered basis, you can work with coordinates. For example, relative to this ordered basis the coordinates (1,3) refer to the vector 1*v1 + 3*v2 = (1,4) (relative to the standard basis (1,0) and (0,1)).

You cannot work with coordinates until you specify an order on the basis vectors. Without an order, you would not be able to tell if the coordinates (1,3) refer to 1*(1,1) + 3*(0,1) or 1*(0,1) + 3*(1,1).

Don't get caught up on the word "ordered". All you need to prove is that your vectors form a basis.

Okay,

So basically then professor says "show that the the vectors v1,v2,v3,v4 is an ordered basis for R4". Then I use the standard definition of a the the basis: and since here there c4 elements(vectors) then the basis has the dimension 4, and if the matrix A which contains the vectors as columns are linear independent then they form a basis for R4?

 

[owlpride]

That's it!