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Basic linear algebra direct sum questions

  1. Nov 29, 2012 #1
    1. The problem statement, all variables and given/known data

    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.



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 29, 2012 #2

    Dick

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    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.
     
  4. Nov 29, 2012 #3
    Yes, I would agree with that since, as I said, reusing the x seems confusing.
     
  5. Nov 29, 2012 #4

    Dick

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    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.
     
  6. Nov 29, 2012 #5
    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.
     
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