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Math Help - Two Abstract Algebra Questions

  1. #1
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    Two Abstract Algebra Questions

    1. Let V be a vector space over a field F, and v_1, v_2,...,v_n \in V. If there is an i>1 such that v_i \in span(v_1, v_2,...,v_{i-1}), then span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n).

    I'm not sure how to start this problem. I don't know what properties to apply to this problem.

    2. Let V be a vector space over a field F, and v_1, v_2,...,v_n \in V. Let u_i = v_i-a_i v_1, where a_i \in F and 2\leq i \leq n. Show that if u_2,...,u_n are linearly dependent, then v_1,...,v_n are also linearly dependent.
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  2. #2
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    Quote Originally Posted by grad444 View Post
    1. Let V be a vector space over a field F, and v_1, v_2,...,v_n \in V. If there is an i>1 such that v_i \in span(v_1, v_2,...,v_{i-1}), then span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n).

    I'm not sure how to start this problem. I don't know what properties to apply to this problem.
    you need to show that span(v_1,...,v_n) \subseteq span(v_1,...,v_{i-1},v_{i+1},...,v_n) and span(v_1,...,v_{i-1},v_{i+1},...,v_n) \subseteq span(v_1,...,v_n). to prove that span(A) \subseteq span(B), you only need to

    show that A \subseteq span(B).


    2. Let V be a vector space over a field F, and v_1, v_2,...,v_n \in V. Let u_i = v_i-a_i v_1, where a_i \in F and 2\leq i \leq n. Show that if u_2,...,u_n are linearly dependent, then v_1,...,v_n

    are also linearly dependent.
    \sum_{i=2}^n c_i u_i =\sum_{i=1}^n c_i v_i, where c_1=-\sum_{i=2}^nc_ia_i. now if u_2, \cdots , u_n are linearly dependent, then there exist c_2, \cdots , c_n, not all 0, such that \sum_{i=2}^n c_i u_i =0. thus ...
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