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Thread: 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 $\displaystyle v_1, v_2,...,v_n \in V$. If there is an i>1 such that $\displaystyle v_i \in span(v_1, v_2,...,v_{i-1})$, then $\displaystyle span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n)$.

    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 $\displaystyle v_1, v_2,...,v_n \in V$. Let $\displaystyle u_i = v_i-a_i v_1$, where $\displaystyle a_i \in F$ and $\displaystyle 2\leq i \leq n$. Show that if $\displaystyle u_2,...,u_n$ are linearly dependent, then $\displaystyle v_1,...,v_n$ 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 $\displaystyle v_1, v_2,...,v_n \in V$. If there is an i>1 such that $\displaystyle v_i \in span(v_1, v_2,...,v_{i-1})$, then $\displaystyle span(v_1,...,v_n)=span(v_1,...,v_{i-1},v_{i+1},...,v_n)$.

    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 $\displaystyle span(v_1,...,v_n) \subseteq span(v_1,...,v_{i-1},v_{i+1},...,v_n)$ and $\displaystyle span(v_1,...,v_{i-1},v_{i+1},...,v_n) \subseteq span(v_1,...,v_n).$ to prove that $\displaystyle span(A) \subseteq span(B),$ you only need to

    show that $\displaystyle A \subseteq span(B).$


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

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