Linear Algebra-Fields and axioms

[mirandasatterley]

Homework Statement

Let F be any field, and fix a є F. Equip the set V = F² with two operations as
follows. Define addition by
(x, y)‡(x', y') := (x + x', y + y' − a), for all x, x', y, y' є F,
and define the scalar multiplication by scalars by
c * (x, y) := (cx, cy − ac + a), for all x, y, c є F.
(i) Prove that F², with these two operations satisfies the two existence axioms for a vector space over F.

The Attempt at a Solution

i) The existence of a zero vector: there exists a zero vector such that 0 + v = v

So I let (x', y') be the zero vector = (0, 0), and got
(x,y) ‡ (0, 0) := (x + 0, y + 0 - a)
= (x, y-a)

At this point, I'm not sure how to deal with the a, or maybe my whole process is wrong! ANy help is appreciated. I'm having the same problem for the other axiom - existence of a negative.

 

[Defennder]

You can't simply designate (0,0) to be the zero vector. The zero vector has to satisfy the rule (x,y) + (zero vector) = (x,y) where + denotes vector addition defined on this vector space.

Instead just denote the zero vector by (w,z) for example, and we have (x,y) + (w,z) = (x,y). Using the defined addition operation, you should be able to deduce what w,z should be to satisfy that. That is then the zero vector.

The same approach works for determinining the scalar multiplicative identity in this field.

 

[mirandasatterley]

So, if (w,z) is the zero vector and (x,y) is any vector, then

(x,y) + (w,z) = (x,y), and addition is defined as
(x,y) ‡ (w,z) := (x+w, y+z-a)

Can I set these equations equal to each other? to get
(x,y) = (x+w, y+z-a) therefore,

x= x+w, and subtracting x from both sides gives w=0, and we also have
y= y+z-a, and solving this gives z=a, so how can I prove that z must be zero?

 

[Almanzo]

The zero vector is a vector which, if added to another vector, gives this other vector as a result.
(x,y)++(0,a) = (x+0,y+a-a) = (x,y).
So (0,a) seems a reasonable zero vector.

If I multiply (x,y) by 0, I get 0(x,y) = (0x,0y-0a+a) = (0,a),
so that's okay, too.

If I multiply (x,y) by 1, I get 1(x,y) = (1x,1y-1a+a) = (x,y), so 1 is a unit element.
Is it also the unique unit element?
Suppose p(x,y) = (x,y). Then (px, whatever) = (x,y), so px = x, so x =1.

 

[Defennder]

z isn't 0. If it were 0, you wouldn't have the zero vector.

 

[mirandasatterley]

For the existence of a negative: there exists a -v such that v + (-v) = 0, does the following proof make sense? Let me know if I am missing something. Thanks.

(x,y) is a vector v
(w,z) is the vector -z, so I need to prove that (w,z) = -(x,y), so I have the equation,

(x,y) + (w,z) = (0,0) from the axiom, and
(x,y) ++ (w,z) := (x+w, y+z-a) from the definition of addition.

therefore (0,0) = (x+w, y+z-a)

And from this, 0=x+w, which solves to w= -x, and
0= y+z-a, which solves to z= a-y

so in the equation for the definition of addition:

(x,y) ++ (-x, a-y) = (x +(-x), y+(a-y) -a) = (0, (y-y) + (a-a)) = (0,0)

 

[mirandasatterley]

z isn't 0. If it were 0, you wouldn't have the zero vector.

I think I understand, so when I solve z=a, and plug this into the addition definition, I am proving that (0,a) is the zero vector since (x,y) ++ (0,a) := (x,y)?

 

[HallsofIvy]

For the existence of a negative: there exists a -v such that v + (-v) = 0, does the following proof make sense? Let me know if I am missing something. Thanks.

(x,y) is a vector v
(w,z) is the vector -z, so I need to prove that (w,z) = -(x,y), so I have the equation,

(x,y) + (w,z) = (0,0) from the axiom, and
(x,y) ++ (w,z) := (x+w, y+z-a) from the definition of addition.

therefore (0,0) = (x+w, y+z-a)

No, no, no. As you were just told, (0,0) is NOT the additive identity for this set!

And from this, 0=x+w, which solves to w= -x, and
0= y+z-a, which solves to z= a-y

so in the equation for the definition of addition:

(x,y) ++ (-x, a-y) = (x +(-x), y+(a-y) -a) = (0, (y-y) + (a-a)) = (0,0)

 

[Defennder]

I think I understand, so when I solve z=a, and plug this into the addition definition, I am proving that (0,a) is the zero vector since (x,y) ++ (0,a) := (x,y)?

Yes, this time you got it.