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Math Help - Basis definition

  1. #1
    Member akhayoon's Avatar
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    Basis definition

    As you know, the definition of a basis is that it is an independent spanning set,

    which means you need two condition for it to be true...

    if U is a subspace of R^n, and B={x1,x2,...,xn}

    1.B must be linearly independent

    2. B is a spanning set of U


    But the thing I don't understand is that how could you have a linear independence for B without having B being a spanning set of U.

    for me this is kind of confusing...mostly since I can't really visualize these types of algebra concepts like i do with calculus.
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  2. #2
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    Quote Originally Posted by akhayoon View Post
    But the thing I don't understand is that how could you have a linear independence for B without having B being a spanning set of U.
    Take B = \{ (0,0,1),(0,1,0) \} for the vector space \mathbb{R}^3. Now B has linear indepedence. But B does not spam the vector space.
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  3. #3
    Member akhayoon's Avatar
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    so this does not span the subspace because there are not enough vectors or whats the reason???

    the book I have really doesn't explain this concept well...
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    Quote Originally Posted by akhayoon View Post
    so this does not span the subspace because there are not enough vectors or whats the reason???
    How about (1,0,0)? Can you express that in terms of (0,1,0) and (0,0,1)?
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  5. #5
    Member akhayoon's Avatar
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    no...but that means that they're independent....

    but why do those two vectors not span R^3...I don't get it


    is there a relationship between the number of vectors and the n of R^n that I can use to help me understand this better...b/c I'm still confused
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    Quote Originally Posted by akhayoon View Post
    but that means that they're independent
    NO. Independent v_1,...,v_j vectors means if k_1v_1+...+k_nv_n = \bold{0} then k_1=k_2=...=k_n=0. They are independent because if k_1(0,1,0)+k_2(0,0,1) = (0,0,0) then (0,k_1,0)+(0,0,k_2) = (0,0,0) then (0,k_1,k_2) = (0,0,0) thus k_1 = k_2 = 0.

    but why do those two vectors not span R^3...I don't get it
    I gave you an example. Consider (1,0,0) it is impossible to express (1,0,0) = k_1(0,1,0)+k_2(0,0,1) because not matter what k_1,k_2 you will never be able to get a non-zero entry in the first coordinate. So it cannot span R^3.
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