# Two Abstract Algebra Questions

• Apr 13th 2009, 08:14 PM
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.
• Apr 14th 2009, 01:36 AM
NonCommAlg
Quote:

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 $A \subseteq span(B).$

Quote:

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 ...