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Math Help - Vector Spaces II

  1. #1
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    Vector Spaces II

    Let V be a vector space over Z_p with dim V = n. How many elements are in V?
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    Quote Originally Posted by Coda202 View Post
    Let V be a vector space over Z_p with dim V = n. How many elements are in V?
     p^n
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    Do you see why that's true. Every vector in the space can be written as a linear combination of vectors in the basis: a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n. There are p possible values for each a_i so by the "fundamental counting principle", there are (p)(p)\cdot\cdot\cdot(p) (n times) so there are p^n possible vectors.
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    Quote Originally Posted by HallsofIvy View Post
    Do you see why that's true. Every vector in the space can be written as a linear combination of vectors in the basis: a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n. There are p possible values for each a_i so by the "fundamental counting principle", there are (p)(p)\cdot\cdot\cdot(p) (n times) so there are p^n possible vectors.
    Though it is evident, but just to make the above post from HallsofIvy more exact -
    Every vector in the space can be written as a linear combination of vectors in the basis in a unique way
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    Quote Originally Posted by aman_cc View Post
    Though it is evident, but just to make the above post from HallsofIvy more exact -
    Every vector in the space can be written as a linear combination of vectors in the basis in a unique way
    I think that what he meant by saying " There are p possible values for each a_{i}"
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  6. #6
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    Quote Originally Posted by Raoh View Post
    I think that what he meant by saying " There are p possible values for each a_{i}"
    Hi Raoh - Sorry I didn't follow your question completely. Let me try to explain what I was trying to emphasize in my post.

    What I meant was that each of the linear combination (and there are p^n linear combination because of the reason you mentioned) is in one-one and onto maping with all the vectors in the vector space. And the reason that happens is only because v1,v2.....vn form a basis. The phrase 'in a unique way' is important to stress the one-one part of the mapping. 'Every' specifies the onto part.
    Last edited by aman_cc; November 30th 2009 at 11:19 PM.
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