Basic linear algebra direct sum questions

[Syrus]

Homework Statement

I'm reading from the first edition of Axler's Linear algebra done right. In the section on sums of vector subspaces, he states:

U = {(x,0,0) ∈ F³ | x ∈ F}

W = {(y,y,0) ∈ F³ | y ∈ F}

and

1.7 U + W = {(x,y,0) ∈ F³ | x,y ∈ F}

However, shouldn't the answer be U + W = {(x+y,y,0) ∈ F³ | x,y ∈ F}? You could write z = x+y, but it seems incorrect to reuse x (instead of z) as a substitute variable in such a case- if this is at all what he's doing.

Homework Equations

The Attempt at a Solution

 

[Dick]

Homework Statement

I'm reading from the first edition of Axler's Linear algebra done right. In the section on sums of vector subspaces, he states:

U = {(x,0,0) ∈ F³ | x ∈ F}

W = {(y,y,0) ∈ F³ | y ∈ F}

and

1.7 U + W = {(x,y,0) ∈ F³ | x,y ∈ F}

However, shouldn't the answer be U + W = {(x+y,y,0) ∈ F³ | x,y ∈ F}? You could write z = x+y, but it seems incorrect to reuse x (instead of z) as a substitute variable in such a case- if this is at all what he's doing.

Homework Equations

The Attempt at a Solution

I'm going to ignore the 1.7. I assume that's a copying artifact. Suppose I told you U + W = {(a,b,0) ∈ F^3 | a,b ∈ F}. Would you agree with that? x and y don't have much meaning outside of the outside of the defining statements.

 

[Syrus]

Yes, I would agree with that since, as I said, reusing the x seems confusing.

 

[Dick]

Yes, I would agree with that since, as I said, reusing the x seems confusing.

Ok, you seem to understand it well enough. And yeah, maybe reusing the symbol is confusing. But {(a,b,0) ∈ F^3 | a,b ∈ F} and {(x,y,0) ∈ F^3 | x,y ∈ F} mean exactly the same thing. Some symbols are 'dummys'. They don't have any meaning outside of the thing they define.

 

[Syrus]

I see what you mean Dick. The symbol x simply represents an arbitrary element of F in each case- not necessarily the same one. I guess it's a matter of preference. This must be what they refer to as developing mathematical maturity, heh heh.