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Homework Help: Trouble understanding vector subspace sum

  1. Aug 2, 2013 #1
    I'm self-studying Linear Algebra and the book I'm using is Linear Algebra done right by Sheldon Axler but I came across something that I don't understand .-
    Suppose [itex]\mathrm U[/itex] is the set of all elements of [itex]\mathbb F ^3[/itex] whose second and third coordinates equal 0, and [itex]\mathrm W[/itex] is the set of all elements of [itex]\mathbb F ^3[/itex] whose first and third coordinates equal 0:
    [tex] \mathrm U = \{ (x , 0 ,0) \in \mathbb F^3: x \in \mathbb F \} \text{ and } \mathrm W= \{(0,y,0) \in \mathbb F^3:y \in F \}[/tex]
    [tex]\mathrm U + \mathrm W = \{ (x, y, 0) : x, y \in \mathbb F \}[/tex]
    As another example, suppose [itex]\mathrm U[/itex] is as above and [itex]\mathrm W[/itex] is the set of all elements of [itex]\mathbb F^3[/itex] whose first and second coordinates equal each other and whose third coordinate equals 0:
    [tex]\mathrm W= \{(y,y,0) \in \mathbb F^3:y \in F \}[/tex]
    Then [itex] \mathrm U + \mathrm W[/itex] is also given by.- (this is the part I don't understand)
    [tex]\mathrm U + \mathrm W = \{ (x, y, 0) : x, y \in \mathbb F \}[/tex]

    why is that it's the same result nevertheless the subspace has changed?
    it shouldn't be something like [tex]\mathrm U + \mathrm W = \{ (x + y, y, 0) : x, y \in \mathbb F \}[/tex] what am I missing here?, thank you very much.
  2. jcsd
  3. Aug 2, 2013 #2


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    The ##x## in the definition of U and the ##x## in the definition of U+W aren't the same. In both cases, U+W is the same subspace of F3, i.e. vectors of the form (x,y,0).
  4. Aug 2, 2013 #3


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    $$\mathrm U + \mathrm W = \{ (x + y, y, 0) : x, y \in \mathbb F \}$$ is the same thing as $$\mathrm U + \mathrm W = \{ (u, v, 0) : u, v \in \mathbb F \}$$ where u=x+y and v=y. Given any u and v you can solve for x and y and vice versa. So they are the same subspace.
  5. Aug 3, 2013 #4
    Thank you very much, now I understand that it just serve as an arbitrary variable to denote all the possible values, it's the first time I read a book as rigorous as this one, but know I get it
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