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Prove that C[a;b] satisfies 8 axioms of vector space

  1. Apr 8, 2014 #1

    sheldonrocks97

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    Gold Member

    1. The problem statement, all variables and given/known data

    Show that C[a; b], with the usual scalar multiplication
    and addition of functions, satis es the eight axioms of a vector space.

    2. Relevant equations

    Eight Axioms of Vector Space:
    A1. x + y = y + z
    A2. (x+y)+z=x+(y+z)
    A3. There exists an element 0 such that x + 0 = 0
    A4. There exists an element -x such that x+(-x)=0
    A5. α(x+y)=αx+αy
    A6. (α+β)x=αx+βx
    A7. (αβ)x=α(βx)
    A8. 1·x=x
    where x, y, and z are all vectors and α, β are scalars

    3. The attempt at a solution

    Let x=<1,2,3> y=<4,5,6> and z=<7,8,9> α=2 and β=3

    1. <1,2,3>+<4,5,6>=<5,7,9> and <4,5,6>+<1,2,3>=<5,7,9>

    Am I on the right track here or is there something I am doing wrong so far?
     
  2. jcsd
  3. Apr 8, 2014 #2

    jbunniii

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    Gold Member

    I'm not sure what you are doing here. Isn't ##C[a; b]## the set of continuous functions on the interval ##[a,b]##? So this has nothing to do with the vectors you have written above. Additionally, even if the space was ##\mathbb{R}^3## instead of ##C[a;b]##, it would not suffice to show that the property is true for a couple of specific vectors. You need to show that the properties hold for general elements of the space.

    To point you in the right direction, start by choosing two arbitrary elements ##x,y \in C[a;b]##. Here, ##x## and ##y## are continuous functions on ##[a,b]##. Now, explain why ##x + y \in C[a;b]##, and why ##x + y = y + x##.
     
  4. Apr 8, 2014 #3

    Mark44

    Staff: Mentor

    C[a, b] is a function space, a kind of vector space where the "vectors" are functions.

    To be more suggestive that we're dealing with functions, I would work with arbitrary elements named, say, f and g, rather than x and y. It doesn't really matter what you call them, but using f and g would remind you more that the elements you're working with are functions.
     
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