Is a Given Set a Subspace Depending on a Parameter?

[Saladsamurai]

Okay then. I just read the section of Axler on subspaces. It says that if U is a subset of V, then to check that U is a subspace of V we need only check that U satisfies the following:

additive identity

0\in U

closed under addition

u,v\in U\text{ implies }u+v\,\in\,U

closed under scalar multiplication

a\in\mathbf{F}\text{ and }u\in U\text{ implies }au\in U

Now I am supposed to use these axioms to verify that for b\in\mathbf{F}, then

{(x_1,x_2,x_3,x_4)\in\mathbf{F}^4:x_3=5x_4+b}

is a subspace of F⁴ iff b=0.

I am not exactly sure how to actually apply those 3 axioms to this problem?

How does one test that u,v\in U\text{ implies }u+v\,\in\,U?

Should I start with something like:

(x1,x2,x3,x4)+(y1,y2,y3,y4)=(x1+y1,x2+y2,5x4+b+5y4+b,x4+y4) ?

 

[Office_Shredder]

(x_1,x_2,x_3,x_4) + (y_1,y_2,y_3,y_4) = (x_1 + y_1, x_2 + y_2, x_3 + y_3, x_4 + y_4)

by definition. So there are two parts to this problem

1) Show if b=0 then this is a subspace
2) Show if this is a subspace then b=0

To work towards 1), if

x_3 = 5x_4 and y_3 = 5y_4

show the RHS satisfies the necessary relationship. Rinse and repeat for the other stuff

To do part 2, you need to find a contradiction. So look at the three things required for the subset to be a subspace, and see if all of them hold

 

[Saladsamurai]

Okay. So just dealing with the x3+y3 part of the addition

x3+y3=5x4+5y4+2b.. I am sorry, but I still do not see why b must equal 0 what part of the definition is being violated if it is not?

 

[snipez90]

Check the additive identity first. What is the additive identity? Now what happens if b is not 0?

 

[Saladsamurai]

Okay, I see now. I forgot about the other properties for a moment.

Can someone help me out wit the formalism of this? If this were a HW problem, my 'proof' would need to follow a certain format. Should I name the proposed subspace and then continue. Like this.

Proposition:
U={(x_1,x_2,x_3,x_4)\in\mathbf{F}^4:x_3=5x_4+b)\text{ is a subspace of }\mathbf{F}^4\text{ iff }b=0 <br />

Proof:

Now I am not sure how to formally state that since a subspace must include the additive identity then b must equal 0. How do you math savvy types do this?

 

[Dick]

You don't have to be that math savvy. You want to solve the equations x3=0, x4=0 and x3=5*x4+b. For what values of b is that possible?

 

[Saladsamurai]

Hey Dick. I realize that I don't have to be math savvy. I want to know what the typical manner in which one presents the verification of a problem such as this.

That way in the future I won't hit snags in proofs with the simple mechanics of presenting it. I will just have to worry about the logic.

 

[Dick]

Well, as I said, present it as a problem of whether the linear equation has the solution (x1,x2,x3,x4)=(0,0,0,0), which is what you need to have the additive identity in the subset.