Prove Linear Algebra Statement: x1E1+...+xnEn=0 then xi=0

[bonfire09]

Homework Statement

Let E₁ = (1, 0, ... ,0), E₂ = (0, 1, 0, ... ,0), ... , E_n = (0, ... ,0, 1)
be the standard unit vectors of Rⁿ. Let x₁ ... ,x_n be numbers. Show that if
x₁E₁+...+x_nE_n=0 then x_i=0 for all i.

Homework Equations

The Attempt at a Solution

Proof By contradiction
Assume to the contrary that x₁E₁+...+x_nE_n=0 then x_i≠0 for some i. We also assume that x₁...x_i-1 and x_i+1...x_n are zero. Rewriting the equation we get
x₁E₁+.x_pE_p+...+x_nE_n=0 where x_pE_p is a nonzero scalar. x_pE_p=-x₁E₁-...-x_pE_p-1-x_pE_p+1-..-x_nE_n. But this leads to a contradiction since we assumed earlier that x₁...x_i-1 and x_i+1...x_n are zero. Thus x₁E₁+...+x_nE_n=0 x_i=0 for all i.

Let me know where my proof begins to fall apart? And how do I go about it?

 

[micromass]

We also assume that x₁...x_i-1 and x_i+1...x_n are zero.

Why can you assume this??

 

[micromass]

Hint: making subscripts like you did in your post is very time-intensive. It is much easier for you to learn LaTeX. I will rewrite the first part of your post using LaTeX here, just push on "QUOTE" to see what I did.

Also see https://www.physicsforums.com/showpost.php?p=3977517&postcount=3 for a guide.

Homework Statement

Let E_1 = (1, 0, ... ,0), E_2 = (0, 1, 0, ... ,0), ... , E_n = (0, ... ,0, 1).
be the standard unit vectors of R^n. Let x_1,...,x_n be numbers. Show that if
x_1E_1 + ... + x_nE_n=0
then x_i=0 for all i.

 

[bonfire09]

Oh thanks. Yeah my proof is bad I just realized I used the conclusion as my assumption.
I don't think I even need to use a proof by contradiction. Isn't just obvious that if one of the scalars is nonzero then the equation is not zero? Wouldn't that suffice as my proof.

 

[micromass]

Oh thanks. Yeah my proof is bad I just realized I used the conclusion as my assumption.
I don't think I even need to use a proof by contradiction. Isn't just obvious that if one of the scalars is nonzero then the equation is not zero? Wouldn't that suffice as my proof.

Well, if your teacher is happy with "it is just obvious", then yes.

If not, try to calculate

x_1E_1+...+x_nE_n

By definition, we know that E_1=(1,0,...). So what is x_1E_1? What is x_2E_2? What happens if you add them?

 

[bonfire09]

Oh x1E1=(x1,...,0) x2E2=(0,x2,...,0) all the way to xnEn=(0,...,xn). So by adding them together you get (x1,x2,...,xn). And they only way to get the zero vector is when x1...xn is zero? Would that be a way to explain it?

 

[micromass]

Oh x1E1=(x1,...,0) x2E2=(0,x2,...,0) all the way to xnEn=(0,...,xn). So by adding them together you get (x1,x2,...,xn). And they only way to get the zero vector is when x1...xn is zero? Would that be a way to explain it?

Yeah, that's what I had in mind.

 

[bonfire09]

alright thanks.

 

[algebrat]

Think about what relations the basis vectors satisfy, if you notice the right thing, the proof is pretty swift.

 

[algebrat]

The problem with the above proof, it doesn't seem to use the fact that the basis is orthonormal. You could potentially "prove something false".

 

[micromass]

The problem with the above proof, it doesn't seem to use the fact that the basis is orthonormal. You could potentially "prove something false".

Why does orthonormal matter?? I really doubt we need it.

 

[algebrat]

Why does orthonormal matter?? I really doubt we need it.

Oops, yeah you're right aren't you. I guess because the slick proof I was thinking of, was take x.e_1=x_1=0. That works right?

So I was just guessing that it relied on some qualities of the basis vector, but maybe the real mistake would be to not refer to the fact that they are linearly independent. We do have to mention that right?

But perhaps the quality of orthogonal was not relavant, as you say.